Question 297092


{{{-6x^2+3x+30}}} Start with the given expression.



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



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



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Looking at the expression {{{2x^2-x-10}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{-1}}}, and the last term is {{{-10}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{-10}}} to get {{{(2)(-10)=-20}}}.



Now the question is: what two whole numbers multiply to {{{-20}}} (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 {{{-20}}} (the previous product).



Factors of {{{-20}}}:

1,2,4,5,10,20

-1,-2,-4,-5,-10,-20



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



These factors pair up and multiply to {{{-20}}}.

1*(-20) = -20
2*(-10) = -20
4*(-5) = -20
(-1)*(20) = -20
(-2)*(10) = -20
(-4)*(5) = -20


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>-20</font></td><td  align="center"><font color=black>1+(-20)=-19</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-10</font></td><td  align="center"><font color=black>2+(-10)=-8</font></td></tr><tr><td  align="center"><font color=red>4</font></td><td  align="center"><font color=red>-5</font></td><td  align="center"><font color=red>4+(-5)=-1</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>20</font></td><td  align="center"><font color=black>-1+20=19</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>10</font></td><td  align="center"><font color=black>-2+10=8</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>-4+5=1</font></td></tr></table>



From the table, we can see that the two numbers {{{4}}} and {{{-5}}} add to {{{-1}}} (the middle coefficient).



So the two numbers {{{4}}} and {{{-5}}} both multiply to {{{-20}}} <font size=4><b>and</b></font> add to {{{-1}}}



Now replace the middle term {{{-1x}}} with {{{4x-5x}}}. Remember, {{{4}}} and {{{-5}}} add to {{{-1}}}. So this shows us that {{{4x-5x=-1x}}}.



{{{2x^2+highlight(4x-5x)-10}}} Replace the second term {{{-1x}}} with {{{4x-5x}}}.



{{{(2x^2+4x)+(-5x-10)}}} Group the terms into two pairs.



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



{{{2x(x+2)-5(x+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.



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



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



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Answer:



So {{{-6x^2+3x+30}}} completely factors to {{{-3(2x-5)(x+2)}}}.



In other words, {{{-6x^2+3x+30=-3(2x-5)(x+2)}}}.



Note: you can check the answer by expanding {{{-3(2x-5)(x+2)}}} to get {{{-6x^2+3x+30}}} or by graphing the original expression and the answer (the two graphs should be identical).