```Question 119790

{{{2(16a^2+2a-3)}}} Factor out the GCF {{{2}}}

Now let's focus on the inner expression {{{16a^2+2a-3}}}

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Looking at {{{16a^2+2a-3}}} we can see that the first term is {{{16a^2}}} and the last term is {{{-3}}} where the coefficients are 16 and -3 respectively.

Now multiply the first coefficient 16 and the last coefficient -3 to get -48. Now what two numbers multiply to -48 and add to the  middle coefficient 2? Let's list all of the factors of -48:

Factors of -48:

1,2,3,4,6,8,12,16,24,48

-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -48

(1)*(-48)

(2)*(-24)

(3)*(-16)

(4)*(-12)

(6)*(-8)

(-1)*(48)

(-2)*(24)

(-3)*(16)

(-4)*(12)

(-6)*(8)

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

Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-48</td><td>1+(-48)=-47</td></tr><tr><td align="center">2</td><td align="center">-24</td><td>2+(-24)=-22</td></tr><tr><td align="center">3</td><td align="center">-16</td><td>3+(-16)=-13</td></tr><tr><td align="center">4</td><td align="center">-12</td><td>4+(-12)=-8</td></tr><tr><td align="center">6</td><td align="center">-8</td><td>6+(-8)=-2</td></tr><tr><td align="center">-1</td><td align="center">48</td><td>-1+48=47</td></tr><tr><td align="center">-2</td><td align="center">24</td><td>-2+24=22</td></tr><tr><td align="center">-3</td><td align="center">16</td><td>-3+16=13</td></tr><tr><td align="center">-4</td><td align="center">12</td><td>-4+12=8</td></tr><tr><td align="center">-6</td><td align="center">8</td><td>-6+8=2</td></tr></table>

From this list we can see that -6 and 8 add up to 2 and multiply to -48

Now looking at the expression {{{16a^2+2a-3}}}, replace {{{2a}}} with {{{-6a+8a}}} (notice {{{-6a+8a}}} adds up to {{{2a}}}. So it is equivalent to {{{2a}}})

{{{16a^2+highlight(-6a+8a)+-3}}}

Now let's factor {{{16a^2-6a+8a-3}}} by grouping:

{{{(16a^2-6a)+(8a-3)}}} Group like terms

{{{2a(8a-3)+1(8a-3)}}} Factor out the GCF of {{{2a}}} out of the first group. Factor out the GCF of {{{1}}} out of the second group

{{{(2a+1)(8a-3)}}} Since we have a common term of {{{8a-3}}}, we can combine like terms

So {{{16a^2-6a+8a-3}}} factors to {{{(2a+1)(8a-3)}}}

So this also means that {{{16a^2+2a-3}}} factors to {{{(2a+1)(8a-3)}}} (since {{{16a^2+2a-3}}} is equivalent to {{{16a^2-6a+8a-3}}})

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{{{2(2a+1)(8a-3)}}} Now reintroduce the GCF {{{2}}}

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