```Question 476853

{{{-(18a^2-17a-15)}}} Factor out the GCF {{{-1}}}.

Now let's try to factor the inner expression {{{18a^2-17a-15}}}

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Looking at the expression {{{18a^2-17a-15}}}, we can see that the first coefficient is {{{18}}}, the second coefficient is {{{-17}}}, and the last term is {{{-15}}}.

Now multiply the first coefficient {{{18}}} by the last term {{{-15}}} to get {{{(18)(-15)=-270}}}.

Now the question is: what two whole numbers multiply to {{{-270}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-17}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-270}}} (the previous product).

Factors of {{{-270}}}:

1,2,3,5,6,9,10,15,18,27,30,45,54,90,135,270

-1,-2,-3,-5,-6,-9,-10,-15,-18,-27,-30,-45,-54,-90,-135,-270

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

These factors pair up and multiply to {{{-270}}}.

1*(-270) = -270
2*(-135) = -270
3*(-90) = -270
5*(-54) = -270
6*(-45) = -270
9*(-30) = -270
10*(-27) = -270
15*(-18) = -270
(-1)*(270) = -270
(-2)*(135) = -270
(-3)*(90) = -270
(-5)*(54) = -270
(-6)*(45) = -270
(-9)*(30) = -270
(-10)*(27) = -270
(-15)*(18) = -270

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-17}}}:

<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>-270</font></td><td  align="center"><font color=black>1+(-270)=-269</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-135</font></td><td  align="center"><font color=black>2+(-135)=-133</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-90</font></td><td  align="center"><font color=black>3+(-90)=-87</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>-54</font></td><td  align="center"><font color=black>5+(-54)=-49</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-45</font></td><td  align="center"><font color=black>6+(-45)=-39</font></td></tr><tr><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>-30</font></td><td  align="center"><font color=black>9+(-30)=-21</font></td></tr><tr><td  align="center"><font color=red>10</font></td><td  align="center"><font color=red>-27</font></td><td  align="center"><font color=red>10+(-27)=-17</font></td></tr><tr><td  align="center"><font color=black>15</font></td><td  align="center"><font color=black>-18</font></td><td  align="center"><font color=black>15+(-18)=-3</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>270</font></td><td  align="center"><font color=black>-1+270=269</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>135</font></td><td  align="center"><font color=black>-2+135=133</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>90</font></td><td  align="center"><font color=black>-3+90=87</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>54</font></td><td  align="center"><font color=black>-5+54=49</font></td></tr><tr><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>45</font></td><td  align="center"><font color=black>-6+45=39</font></td></tr><tr><td  align="center"><font color=black>-9</font></td><td  align="center"><font color=black>30</font></td><td  align="center"><font color=black>-9+30=21</font></td></tr><tr><td  align="center"><font color=black>-10</font></td><td  align="center"><font color=black>27</font></td><td  align="center"><font color=black>-10+27=17</font></td></tr><tr><td  align="center"><font color=black>-15</font></td><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>-15+18=3</font></td></tr></table>

From the table, we can see that the two numbers {{{10}}} and {{{-27}}} add to {{{-17}}} (the middle coefficient).

So the two numbers {{{10}}} and {{{-27}}} both multiply to {{{-270}}} <font size=4><b>and</b></font> add to {{{-17}}}

Now replace the middle term {{{-17a}}} with {{{10a-27a}}}. Remember, {{{10}}} and {{{-27}}} add to {{{-17}}}. So this shows us that {{{10a-27a=-17a}}}.

{{{18a^2+highlight(10a-27a)-15}}} Replace the second term {{{-17a}}} with {{{10a-27a}}}.

{{{(18a^2+10a)+(-27a-15)}}} Group the terms into two pairs.

{{{2a(9a+5)+(-27a-15)}}} Factor out the GCF {{{2a}}} from the first group.

{{{2a(9a+5)-3(9a+5)}}} 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.

{{{(2a-3)(9a+5)}}} Combine like terms. Or factor out the common term {{{9a+5}}}

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So {{{-1(18a^2-17a-15)}}} then factors further to {{{-(2a-3)(9a+5)}}}

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