```Question 166508

Looking at the expression {{{3x^2+2x-8}}}, we can see that the first coefficient is {{{3}}}, the second coefficient is {{{2}}}, and the last term is {{{-8}}}.

Now multiply the first coefficient {{{3}}} by the last term {{{-8}}} to get {{{(3)(-8)=-24}}}.

Now the question is: what two whole numbers multiply to {{{-24}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{2}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-24}}} (the previous product).

Factors of {{{-24}}}:

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

-1,-2,-3,-4,-6,-8,-12,-24

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to {{{-24}}}.

1*(-24)
2*(-12)
3*(-8)
4*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{2}}}:

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-24</font></td><td  align="center"><font color=black>1+(-24)=-23</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>2+(-12)=-10</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>3+(-8)=-5</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>4+(-6)=-2</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>24</font></td><td  align="center"><font color=black>-1+24=23</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-2+12=10</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>-3+8=5</font></td></tr><tr><td  align="center"><font color=red>-4</font></td><td  align="center"><font color=red>6</font></td><td  align="center"><font color=red>-4+6=2</font></td></tr></table>

From the table, we can see that the two numbers {{{-4}}} and {{{6}}} add to {{{2}}} (the middle coefficient).

So the two numbers {{{-4}}} and {{{6}}} both multiply to {{{-24}}} <font size=4><b>and</b></font> add to {{{2}}}

Now replace the middle term {{{2x}}} with {{{-4x+6x}}}. Remember, {{{-4}}} and {{{6}}} add to {{{2}}}. So this shows us that {{{-4x+6x=2x}}}.

{{{3x^2+highlight(-4x+6x)-8}}} Replace the second term {{{2x}}} with {{{-4x+6x}}}.

{{{(3x^2-4x)+(6x-8)}}} Group the terms into two pairs.

{{{x(3x-4)+(6x-8)}}} Factor out the GCF {{{x}}} from the first group.

{{{x(3x-4)+2(3x-4)}}} Factor out {{{2}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

{{{(x+2)(3x-4)}}} Combine like terms. Or factor out the common term {{{3x-4}}}

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