Question 704242


{{{-15x^2-81x-30}}} Start with the given expression.



{{{-3(5x^2+27x+10)}}} Factor out the GCF {{{-3}}}.



Now let's try to factor the inner expression {{{5x^2+27x+10}}}



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Looking at the expression {{{5x^2+27x+10}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{27}}}, and the last term is {{{10}}}.



Now multiply the first coefficient {{{5}}} by the last term {{{10}}} to get {{{(5)(10)=50}}}.



Now the question is: what two whole numbers multiply to {{{50}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{27}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{50}}} (the previous product).



Factors of {{{50}}}:

1,2,5,10,25,50

-1,-2,-5,-10,-25,-50



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



These factors pair up and multiply to {{{50}}}.

1*50 = 50
2*25 = 50
5*10 = 50
(-1)*(-50) = 50
(-2)*(-25) = 50
(-5)*(-10) = 50


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{27}}}:



<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>50</font></td><td  align="center"><font color=black>1+50=51</font></td></tr><tr><td  align="center"><font color=red>2</font></td><td  align="center"><font color=red>25</font></td><td  align="center"><font color=red>2+25=27</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>10</font></td><td  align="center"><font color=black>5+10=15</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-50</font></td><td  align="center"><font color=black>-1+(-50)=-51</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-25</font></td><td  align="center"><font color=black>-2+(-25)=-27</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>-10</font></td><td  align="center"><font color=black>-5+(-10)=-15</font></td></tr></table>



From the table, we can see that the two numbers {{{2}}} and {{{25}}} add to {{{27}}} (the middle coefficient).



So the two numbers {{{2}}} and {{{25}}} both multiply to {{{50}}} <font size=4><b>and</b></font> add to {{{27}}}



Now replace the middle term {{{27x}}} with {{{2x+25x}}}. Remember, {{{2}}} and {{{25}}} add to {{{27}}}. So this shows us that {{{2x+25x=27x}}}.



{{{5x^2+highlight(2x+25x)+10}}} Replace the second term {{{27x}}} with {{{2x+25x}}}.



{{{(5x^2+2x)+(25x+10)}}} Group the terms into two pairs.



{{{x(5x+2)+(25x+10)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(5x+2)+5(5x+2)}}} Factor out {{{5}}} 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+5)(5x+2)}}} Combine like terms. Or factor out the common term {{{5x+2}}}



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So {{{-3(5x^2+27x+10)}}} then factors further to {{{-3(x+5)(5x+2)}}}



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Answer:



So {{{-15x^2-81x-30}}} completely factors to {{{-3(x+5)(5x+2)}}}.



In other words, {{{-15x^2-81x-30=-3(x+5)(5x+2)}}}.



Note: you can check the answer by expanding {{{-3(x+5)(5x+2)}}} to get {{{-15x^2-81x-30}}} or by graphing the original expression and the answer (the two graphs should be identical).