Question 243968
{{{3x^2+19x-2x-6}}} Start with the given expression.



{{{3x^2+17x-6}}} Combine like terms.



Looking at the expression {{{3x^2+17x-6}}}, we can see that the first coefficient is {{{3}}}, the second coefficient is {{{17}}}, and the last term is {{{-6}}}.



Now multiply the first coefficient {{{3}}} by the last term {{{-6}}} to get {{{(3)(-6)=-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 {{{17}}}?



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 {{{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>-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><tr><td  align="center"><font color=red>-1</font></td><td  align="center"><font color=red>18</font></td><td  align="center"><font color=red>-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 {{{-1}}} and {{{18}}} add to {{{17}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{18}}} both multiply to {{{-18}}} <font size=4><b>and</b></font> add to {{{17}}}



Now replace the middle term {{{17x}}} with {{{-x+18x}}}. Remember, {{{-1}}} and {{{18}}} add to {{{17}}}. So this shows us that {{{-x+18x=17x}}}.



{{{3x^2+highlight(-x+18x)-6}}} Replace the second term {{{17x}}} with {{{-x+18x}}}.



{{{(3x^2-x)+(18x-6)}}} Group the terms into two pairs.



{{{x(3x-1)+(18x-6)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(3x-1)+6(3x-1)}}} Factor out {{{6}}} 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+6)(3x-1)}}} Combine like terms. Or factor out the common term {{{3x-1}}}


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Answer:



So {{{3x^2+19x-2x-6}}} factors to {{{(x+6)(3x-1)}}}.



In other words, {{{3x^2+19x-2x-6=(x+6)(3x-1)}}}



Note: you can check the answer by FOILing {{{(x+6)(3x-1)}}} to get {{{3x^2+17x-6}}} or by graphing the original expression and the answer (the two graphs should be identical).