<PRE><B>Question 171535</B>
I&#39;ll do the first three to get you started



# 1





{{{12x^2+6x+18}}} Start with the given expression



{{{6(2x^2+x+3)}}} Factor out the GCF {{{6}}}



Now let&#39;s focus on the inner expression {{{2x^2+x+3}}}





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Looking at {{{2x^2+1x+3}}} we can see that the first term is {{{2x^2}}} and the last term is {{{3}}} where the coefficients are 2 and 3 respectively.


Now multiply the first coefficient 2 and the last coefficient 3 to get 6. Now what two numbers multiply to 6 and add to the  middle coefficient 1? 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 two negative numbers multiplied together make a positive number



Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1


&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=7&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=5&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)=-7&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)=-5&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;

None of these pairs of factors add to 1. So the expression {{{2x^2+x+3}}} cannot be factored



So {{{6(2x^2+x+3)}}} just remains as {{{6(2x^2+x+3)}}}


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Answer:


So {{{12x^2+6x+18}}} factors to {{{6(2x^2+x+3)}}}

    



&lt;hr&gt;



# 2





Looking at {{{1m^2+11m+18}}} we can see that the first term is {{{1m^2}}} and the last term is {{{18}}} where the coefficients are 1 and 18 respectively.


Now multiply the first coefficient 1 and the last coefficient 18 to get 18. Now what two numbers multiply to 18 and add to the  middle coefficient 11? 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 two negative numbers multiplied together make a positive number



Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11


&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=19&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=11&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=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;-18&lt;/td&gt;&lt;td&gt;-1+(-18)=-19&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)=-11&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)=-9&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From this list we can see that 2 and 9 add up to 11 and multiply to 18



Now looking at the expression {{{1m^2+11m+18}}}, replace {{{11m}}} with {{{2m+9m}}} (notice {{{2m+9m}}} adds up to {{{11m}}}. So it is equivalent to {{{11m}}})


{{{1m^2+highlight(2m+9m)+18}}}



Now let&#39;s factor {{{1m^2+2m+9m+18}}} by grouping:



{{{(1m^2+2m)+(9m+18)}}} Group like terms



{{{m(m+2)+9(m+2)}}} Factor out the GCF of {{{m}}} out of the first group. Factor out the GCF of {{{9}}} out of the second group



{{{(m+9)(m+2)}}} Since we have a common term of {{{m+2}}}, we can combine like terms


So {{{1m^2+2m+9m+18}}} factors to {{{(m+9)(m+2)}}}



So this also means that {{{1m^2+11m+18}}} factors to {{{(m+9)(m+2)}}} (since {{{1m^2+11m+18}}} is equivalent to {{{1m^2+2m+9m+18}}})




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     Answer:

So {{{m^2+11m+18}}} factors to {{{(m+9)(m+2)}}}





&lt;hr&gt;



# 3




Looking at {{{1x^2-14x-15}}} we can see that the first term is {{{1x^2}}} and the last term is {{{-15}}} where the coefficients are 1 and -15 respectively.


Now multiply the first coefficient 1 and the last coefficient -15 to get -15. Now what two numbers multiply to -15 and add to the  middle coefficient -14? Let&#39;s list all of the factors of -15:




Factors of -15:

1,3,5,15


-1,-3,-5,-15 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -15

(1)*(-15)

(3)*(-5)

(-1)*(15)

(-3)*(5)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to -14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -14


&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;-15&lt;/td&gt;&lt;td&gt;1+(-15)=-14&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;-5&lt;/td&gt;&lt;td&gt;3+(-5)=-2&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;15&lt;/td&gt;&lt;td&gt;-1+15=14&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;5&lt;/td&gt;&lt;td&gt;-3+5=2&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From this list we can see that 1 and -15 add up to -14 and multiply to -15



Now looking at the expression {{{1x^2-14x-15}}}, replace {{{-14x}}} with {{{1x+-15x}}} (notice {{{1x+-15x}}} adds up to {{{-14x}}}. So it is equivalent to {{{-14x}}})


{{{1x^2+highlight(1x+-15x)+-15}}}



Now let&#39;s factor {{{1x^2+1x-15x-15}}} by grouping:



{{{(1x^2+1x)+(-15x-15)}}} Group like terms



{{{x(x+1)-15(x+1)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-15}}} out of the second group



{{{(x-15)(x+1)}}} Since we have a common term of {{{x+1}}}, we can combine like terms


So {{{1x^2+1x-15x-15}}} factors to {{{(x-15)(x+1)}}}



So this also means that {{{1x^2-14x-15}}} factors to {{{(x-15)(x+1)}}} (since {{{1x^2-14x-15}}} is equivalent to {{{1x^2+1x-15x-15}}})




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     Answer:

So {{{x^2-14x-15}}} factors to {{{(x-15)(x+1)}}}
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