```Question 249349

Looking at the expression {{{6y^2-5y-6}}}, we can see that the first coefficient is {{{6}}}, the second coefficient is {{{-5}}}, and the last term is {{{-6}}}.

Now multiply the first coefficient {{{6}}} by the last term {{{-6}}} to get {{{(6)(-6)=-36}}}.

Now the question is: what two whole numbers multiply to {{{-36}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-5}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-36}}} (the previous product).

Factors of {{{-36}}}:

1,2,3,4,6,9,12,18,36

-1,-2,-3,-4,-6,-9,-12,-18,-36

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

These factors pair up and multiply to {{{-36}}}.

1*(-36) = -36
2*(-18) = -36
3*(-12) = -36
4*(-9) = -36
6*(-6) = -36
(-1)*(36) = -36
(-2)*(18) = -36
(-3)*(12) = -36
(-4)*(9) = -36
(-6)*(6) = -36

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-5}}}:

<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>-36</font></td><td  align="center"><font color=black>1+(-36)=-35</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-18</font></td><td  align="center"><font color=black>2+(-18)=-16</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>3+(-12)=-9</font></td></tr><tr><td  align="center"><font color=red>4</font></td><td  align="center"><font color=red>-9</font></td><td  align="center"><font color=red>4+(-9)=-5</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>6+(-6)=0</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>36</font></td><td  align="center"><font color=black>-1+36=35</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>-2+18=16</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-3+12=9</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>-4+9=5</font></td></tr><tr><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-6+6=0</font></td></tr></table>

From the table, we can see that the two numbers {{{4}}} and {{{-9}}} add to {{{-5}}} (the middle coefficient).

So the two numbers {{{4}}} and {{{-9}}} both multiply to {{{-36}}} <font size=4><b>and</b></font> add to {{{-5}}}

Now replace the middle term {{{-5y}}} with {{{4y-9y}}}. Remember, {{{4}}} and {{{-9}}} add to {{{-5}}}. So this shows us that {{{4y-9y=-5y}}}.

{{{6y^2+highlight(4y-9y)-6}}} Replace the second term {{{-5y}}} with {{{4y-9y}}}.

{{{(6y^2+4y)+(-9y-6)}}} Group the terms into two pairs.

{{{2y(3y+2)+(-9y-6)}}} Factor out the GCF {{{2y}}} from the first group.

{{{2y(3y+2)-3(3y+2)}}} Factor out {{{3}}} 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.

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

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