```Question 547991

Looking at the expression {{{5p^2-p-18}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{-1}}}, and the last term is {{{-18}}}.

Now multiply the first coefficient {{{5}}} by the last term {{{-18}}} to get {{{(5)(-18)=-90}}}.

Now the question is: what two whole numbers multiply to {{{-90}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-1}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-90}}} (the previous product).

Factors of {{{-90}}}:

1,2,3,5,6,9,10,15,18,30,45,90

-1,-2,-3,-5,-6,-9,-10,-15,-18,-30,-45,-90

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

These factors pair up and multiply to {{{-90}}}.

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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-1}}}:

<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>-90</font></td><td  align="center"><font color=black>1+(-90)=-89</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-45</font></td><td  align="center"><font color=black>2+(-45)=-43</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-30</font></td><td  align="center"><font color=black>3+(-30)=-27</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>-18</font></td><td  align="center"><font color=black>5+(-18)=-13</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-15</font></td><td  align="center"><font color=black>6+(-15)=-9</font></td></tr><tr><td  align="center"><font color=red>9</font></td><td  align="center"><font color=red>-10</font></td><td  align="center"><font color=red>9+(-10)=-1</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>90</font></td><td  align="center"><font color=black>-1+90=89</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>45</font></td><td  align="center"><font color=black>-2+45=43</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>30</font></td><td  align="center"><font color=black>-3+30=27</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>-5+18=13</font></td></tr><tr><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>15</font></td><td  align="center"><font color=black>-6+15=9</font></td></tr><tr><td  align="center"><font color=black>-9</font></td><td  align="center"><font color=black>10</font></td><td  align="center"><font color=black>-9+10=1</font></td></tr></table>

From the table, we can see that the two numbers {{{9}}} and {{{-10}}} add to {{{-1}}} (the middle coefficient).

So the two numbers {{{9}}} and {{{-10}}} both multiply to {{{-90}}} <font size=4><b>and</b></font> add to {{{-1}}}

Now replace the middle term {{{-1p}}} with {{{9p-10p}}}. Remember, {{{9}}} and {{{-10}}} add to {{{-1}}}. So this shows us that {{{9p-10p=-1p}}}.

{{{5p^2+highlight(9p-10p)-18}}} Replace the second term {{{-1p}}} with {{{9p-10p}}}.

{{{(5p^2+9p)+(-10p-18)}}} Group the terms into two pairs.

{{{p(5p+9)+(-10p-18)}}} Factor out the GCF {{{p}}} from the first group.

{{{p(5p+9)-2(5p+9)}}} 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.

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

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So {{{5p^2-p-18}}} factors to {{{(p-2)(5p+9)}}}.

In other words, {{{5p^2-p-18=(p-2)(5p+9)}}}.

Note: you can check the answer by expanding {{{(p-2)(5p+9)}}} to get {{{5p^2-p-18}}} or by graphing the original expression and the answer (the two graphs should be identical).
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