Question 182957
I'm assuming that you want to factor.

{{{5x^2(x^2-7x-18)}}} Factor out the GCF {{{5x^2}}}

Now let's focus on the inner expression {{{x^2-7x-18}}}

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Looking at the expression {{{x^2-7x-18}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-7}}}, and the last term is {{{-18}}}.

Now multiply the first coefficient {{{1}}} by the last term {{{-18}}} to get {{{(1)(-18)=-18}}}.

Now the question is: what two whole numbers multiply to {{{-18}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-7}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-18}}} (the previous product).

Factors of {{{-18}}}:

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18

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

These factors pair up and multiply to {{{-18}}}.

1*(-18)
2*(-9)
3*(-6)
(-1)*(18)
(-2)*(9)
(-3)*(6)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-7}}}:

<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>-18</font></td><td  align="center"><font color=black>1+(-18)=-17</font></td></tr><tr><td  align="center"><font color=red>2</font></td><td  align="center"><font color=red>-9</font></td><td  align="center"><font color=red>2+(-9)=-7</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>3+(-6)=-3</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>-1+18=17</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>-2+9=7</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-3+6=3</font></td></tr></table>

From the table, we can see that the two numbers {{{2}}} and {{{-9}}} add to {{{-7}}} (the middle coefficient).

So the two numbers {{{2}}} and {{{-9}}} both multiply to {{{-18}}} <font size=4><b>and</b></font> add to {{{-7}}}

Now replace the middle term {{{-7x}}} with {{{2x-9x}}}. Remember, {{{2}}} and {{{-9}}} add to {{{-7}}}. So this shows us that {{{2x-9x=-7x}}}.

{{{x^2+highlight(2x-9x)-18}}} Replace the second term {{{-7x}}} with {{{2x-9x}}}.

{{{(x^2+2x)+(-9x-18)}}} Group the terms into two pairs.

{{{x(x+2)+(-9x-18)}}} Factor out the GCF {{{x}}} from the first group.

{{{x(x+2)-9(x+2)}}} Factor out {{{9}}} 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-9)(x+2)}}} Combine like terms. Or factor out the common term {{{x+2}}}

So {{{x^2-7x-18}}} factors to {{{(x-9)(x+2)}}}.

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So our expression goes from {{{5x^2(x^2-7x-18)}}} and factors further to {{{5x^2(x-9)(x+2)}}}

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