<PRE><B>Question 117667</B>
#1





Looking at {{{4x^2-12x+5}}} we can see that the first term is {{{4x^2}}} and the last term is {{{5}}} where the coefficients are 4 and 5 respectively.


Now multiply the first coefficient 4 and the last coefficient 5 to get 20. Now what two numbers multiply to 20 and add to the  middle coefficient -12? Let&#39;s list all of the factors of 20:




Factors of 20:

1,2,4,5,10,20


-1,-2,-4,-5,-10,-20 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 20

1*20

2*10

4*5

(-1)*(-20)

(-2)*(-10)

(-4)*(-5)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12


&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;20&lt;/td&gt;&lt;td&gt;1+20=21&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;10&lt;/td&gt;&lt;td&gt;2+10=12&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;4&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;5&lt;/td&gt;&lt;td&gt;4+5=9&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-20&lt;/td&gt;&lt;td&gt;-1+(-20)=-21&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-10&lt;/td&gt;&lt;td&gt;-2+(-10)=-12&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-4&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-5&lt;/td&gt;&lt;td&gt;-4+(-5)=-9&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From this list we can see that -2 and -10 add up to -12 and multiply to 20



Now looking at the expression {{{4x^2-12x+5}}}, replace {{{-12x}}} with {{{-2x+-10x}}} (notice {{{-2x+-10x}}} adds up to {{{-12x}}}. So it is equivalent to {{{-12x}}})


{{{4x^2+highlight(-2x+-10x)+5}}}



Now let&#39;s factor {{{4x^2-2x-10x+5}}} by grouping:



{{{(4x^2-2x)+(-10x+5)}}} Group like terms



{{{2x(2x-1)-5(2x-1)}}} Factor out the GCF of {{{2x}}} out of the first group. Factor out the GCF of {{{-5}}} out of the second group



{{{(2x-5)(2x-1)}}} Since we have a common term of {{{2x-1}}}, we can combine like terms


So {{{4x^2-2x-10x+5}}} factors to {{{(2x-5)(2x-1)}}}



So this also means that {{{4x^2-12x+5}}} factors to {{{(2x-5)(2x-1)}}} (since {{{4x^2-12x+5}}} is equivalent to {{{4x^2-2x-10x+5}}})


-------------------------------

Answer:


So {{{4x^2-12x+5}}} factors to {{{(2x-5)(2x-1)}}}




&lt;hr&gt;



#2





Looking at {{{3x^2-5x-2}}} we can see that the first term is {{{3x^2}}} and the last term is {{{-2}}} where the coefficients are 3 and -2 respectively.


Now multiply the first coefficient 3 and the last coefficient -2 to get -6. Now what two numbers multiply to -6 and add to the  middle coefficient -5? Let&#39;s list all of the factors of -6:




Factors of -6:

1,2,3,6


-1,-2,-3,-6 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -6

(1)*(-6)

(2)*(-3)

(-1)*(6)

(-2)*(3)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5


&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-6&lt;/td&gt;&lt;td&gt;1+(-6)=-5&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-3&lt;/td&gt;&lt;td&gt;2+(-3)=-1&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;6&lt;/td&gt;&lt;td&gt;-1+6=5&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;3&lt;/td&gt;&lt;td&gt;-2+3=1&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From this list we can see that 1 and -6 add up to -5 and multiply to -6



Now looking at the expression {{{3x^2-5x-2}}}, replace {{{-5x}}} with {{{1x+-6x}}} (notice {{{1x+-6x}}} adds up to {{{-5x}}}. So it is equivalent to {{{-5x}}})


{{{3x^2+highlight(1x+-6x)+-2}}}



Now let&#39;s factor {{{3x^2+1x-6x-2}}} by grouping:



{{{(3x^2+1x)+(-6x-2)}}} Group like terms



{{{x(3x+1)-2(3x+1)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-2}}} out of the second group



{{{(x-2)(3x+1)}}} Since we have a common term of {{{3x+1}}}, we can combine like terms


So {{{3x^2+1x-6x-2}}} factors to {{{(x-2)(3x+1)}}}



So this also means that {{{3x^2-5x-2}}} factors to {{{(x-2)(3x+1)}}} (since {{{3x^2-5x-2}}} is equivalent to {{{3x^2+1x-6x-2}}})


-------------------------------

Answer:


So {{{3x^2-5x-2}}} factors to {{{(x-2)(3x+1)}}}




&lt;hr&gt;



#3





Looking at {{{3y^2+7y-6}}} we can see that the first term is {{{3y^2}}} and the last term is {{{-6}}} where the coefficients are 3 and -6 respectively.


Now multiply the first coefficient 3 and the last coefficient -6 to get -18. Now what two numbers multiply to -18 and add to the  middle coefficient 7? Let&#39;s list all of the factors of -18:




Factors of -18:

1,2,3,6,9,18


-1,-2,-3,-6,-9,-18 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -18

(1)*(-18)

(2)*(-9)

(3)*(-6)

(-1)*(18)

(-2)*(9)

(-3)*(6)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7


&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-18&lt;/td&gt;&lt;td&gt;1+(-18)=-17&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-9&lt;/td&gt;&lt;td&gt;2+(-9)=-7&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;3&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-6&lt;/td&gt;&lt;td&gt;3+(-6)=-3&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;18&lt;/td&gt;&lt;td&gt;-1+18=17&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;9&lt;/td&gt;&lt;td&gt;-2+9=7&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-3&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;6&lt;/td&gt;&lt;td&gt;-3+6=3&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From this list we can see that -2 and 9 add up to 7 and multiply to -18



Now looking at the expression {{{3y^2+7y-6}}}, replace {{{7y}}} with {{{-2y+9y}}} (notice {{{-2y+9y}}} adds up to {{{7y}}}. So it is equivalent to {{{7y}}})


{{{3y^2+highlight(-2y+9y)+-6}}}



Now let&#39;s factor {{{3y^2-2y+9y-6}}} by grouping:



{{{(3y^2-2y)+(9y-6)}}} Group like terms



{{{y(3y-2)+3(3y-2)}}} Factor out the GCF of {{{y}}} out of the first group. Factor out the GCF of {{{3}}} out of the second group



{{{(y+3)(3y-2)}}} Since we have a common term of {{{3y-2}}}, we can combine like terms


So {{{3y^2-2y+9y-6}}} factors to {{{(y+3)(3y-2)}}}



So this also means that {{{3y^2+7y-6}}} factors to {{{(y+3)(3y-2)}}} (since {{{3y^2+7y-6}}} is equivalent to {{{3y^2-2y+9y-6}}})


-------------------------------

Answer:


So {{{3y^2+7y-6}}} factors to {{{(y+3)(3y-2)}}}

</PRE>