Question 131011
# 1





{{{3a^2+12ab+12b^2}}} Start with the given expression



{{{3(a^2+4ab+4b^2)}}} Factor out the GCF {{{3}}}



Now let's focus on the inner expression {{{a^2+4ab+4b^2}}}





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Looking at {{{1a^2+4ab+4b^2}}} we can see that the first term is {{{1a^2}}} and the last term is {{{4b^2}}} where the coefficients are 1 and 4 respectively.


Now multiply the first coefficient 1 and the last coefficient 4 to get 4. Now what two numbers multiply to 4 and add to the  middle coefficient 4? Let's list all of the factors of 4:




Factors of 4:

1,2


-1,-2 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 4

1*4

2*2

(-1)*(-4)

(-2)*(-2)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">4</td><td>1+4=5</td></tr><tr><td align="center">2</td><td align="center">2</td><td>2+2=4</td></tr><tr><td align="center">-1</td><td align="center">-4</td><td>-1+(-4)=-5</td></tr><tr><td align="center">-2</td><td align="center">-2</td><td>-2+(-2)=-4</td></tr></table>



From this list we can see that 2 and 2 add up to 4 and multiply to 4



Now looking at the expression {{{1a^2+4ab+4b^2}}}, replace {{{4ab}}} with {{{2ab+2ab}}} (notice {{{2ab+2ab}}} adds up to {{{4ab}}}. So it is equivalent to {{{4ab}}})


{{{1a^2+highlight(2ab+2ab)+4b^2}}}



Now let's factor {{{1a^2+2ab+2ab+4b^2}}} by grouping:



{{{(1a^2+2ab)+(2ab+4b^2)}}} Group like terms



{{{a(a+2b)+2b(a+2b)}}} Factor out the GCF of {{{a}}} out of the first group. Factor out the GCF of {{{2b}}} out of the second group



{{{(a+2b)(a+2b)}}} Since we have a common term of {{{a+2b}}}, we can combine like terms


So {{{1a^2+2ab+2ab+4b^2}}} factors to {{{(a+2b)(a+2b)}}}



So this also means that {{{1a^2+4ab+4b^2}}} factors to {{{(a+2b)(a+2b)}}} (since {{{1a^2+4ab+4b^2}}} is equivalent to {{{1a^2+2ab+2ab+4b^2}}})



note:  {{{(a+2b)(a+2b)}}} is equivalent to  {{{(a+2b)^2}}} since the term {{{a+2b}}} occurs twice. So {{{1a^2+4ab+4b^2}}} also factors to {{{(a+2b)^2}}}




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So our expression goes from {{{3(a^2+4ab+4b^2)}}} and factors further to {{{3(a+2b)^2}}}



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Answer:


So {{{3a^2+12ab+12b^2}}} factors to {{{3(a+2b)^2}}}

    




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# 2





Looking at {{{8x^2-26x-15}}} we can see that the first term is {{{8x^2}}} and the last term is {{{-15}}} where the coefficients are 8 and -15 respectively.


Now multiply the first coefficient 8 and the last coefficient -15 to get -120. Now what two numbers multiply to -120 and add to the  middle coefficient -26? Let's list all of the factors of -120:




Factors of -120:

1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120


-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -120

(1)*(-120)

(2)*(-60)

(3)*(-40)

(4)*(-30)

(5)*(-24)

(6)*(-20)

(8)*(-15)

(10)*(-12)

(-1)*(120)

(-2)*(60)

(-3)*(40)

(-4)*(30)

(-5)*(24)

(-6)*(20)

(-8)*(15)

(-10)*(12)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to -26? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -26


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-120</td><td>1+(-120)=-119</td></tr><tr><td align="center">2</td><td align="center">-60</td><td>2+(-60)=-58</td></tr><tr><td align="center">3</td><td align="center">-40</td><td>3+(-40)=-37</td></tr><tr><td align="center">4</td><td align="center">-30</td><td>4+(-30)=-26</td></tr><tr><td align="center">5</td><td align="center">-24</td><td>5+(-24)=-19</td></tr><tr><td align="center">6</td><td align="center">-20</td><td>6+(-20)=-14</td></tr><tr><td align="center">8</td><td align="center">-15</td><td>8+(-15)=-7</td></tr><tr><td align="center">10</td><td align="center">-12</td><td>10+(-12)=-2</td></tr><tr><td align="center">-1</td><td align="center">120</td><td>-1+120=119</td></tr><tr><td align="center">-2</td><td align="center">60</td><td>-2+60=58</td></tr><tr><td align="center">-3</td><td align="center">40</td><td>-3+40=37</td></tr><tr><td align="center">-4</td><td align="center">30</td><td>-4+30=26</td></tr><tr><td align="center">-5</td><td align="center">24</td><td>-5+24=19</td></tr><tr><td align="center">-6</td><td align="center">20</td><td>-6+20=14</td></tr><tr><td align="center">-8</td><td align="center">15</td><td>-8+15=7</td></tr><tr><td align="center">-10</td><td align="center">12</td><td>-10+12=2</td></tr></table>



From this list we can see that 4 and -30 add up to -26 and multiply to -120



Now looking at the expression {{{8x^2-26x-15}}}, replace {{{-26x}}} with {{{4x+-30x}}} (notice {{{4x+-30x}}} adds up to {{{-26x}}}. So it is equivalent to {{{-26x}}})


{{{8x^2+highlight(4x+-30x)+-15}}}



Now let's factor {{{8x^2+4x-30x-15}}} by grouping:



{{{(8x^2+4x)+(-30x-15)}}} Group like terms



{{{4x(2x+1)-15(2x+1)}}} Factor out the GCF of {{{4x}}} out of the first group. Factor out the GCF of {{{-15}}} out of the second group



{{{(4x-15)(2x+1)}}} Since we have a common term of {{{2x+1}}}, we can combine like terms


So {{{8x^2+4x-30x-15}}} factors to {{{(4x-15)(2x+1)}}}



So this also means that {{{8x^2-26x-15}}} factors to {{{(4x-15)(2x+1)}}} (since {{{8x^2-26x-15}}} is equivalent to {{{8x^2+4x-30x-15}}})




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     Answer:

So {{{8x^2-26x-15}}} factors to {{{(4x-15)(2x+1)}}}