Question 158076
I'm assuming that you want to factor?



{{{2c^2+10c-48}}} Start with the given expression



{{{2(c^2+5c-24)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{c^2+5c-24}}}





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Looking at the expression {{{c^2+5c-24}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{5}}}, and the last term is {{{-24}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-24}}} to get {{{(1)(-24)=-24}}}.



Now the question is: what two whole numbers multiply to {{{-24}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{5}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-24}}} (the previous product).



Factors of {{{-24}}}:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24



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



These factors pair up and multiply to {{{-24}}}.

1*(-24)
2*(-12)
3*(-8)
4*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{5}}}:



<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>-24</font></td><td  align="center"><font color=black>1+(-24)=-23</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>2+(-12)=-10</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>3+(-8)=-5</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>4+(-6)=-2</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>24</font></td><td  align="center"><font color=black>-1+24=23</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-2+12=10</font></td></tr><tr><td  align="center"><font color=red>-3</font></td><td  align="center"><font color=red>8</font></td><td  align="center"><font color=red>-3+8=5</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-4+6=2</font></td></tr></table>



From the table, we can see that the two numbers {{{-3}}} and {{{8}}} add to {{{5}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{8}}} both multiply to {{{-24}}} <font size=4><b>and</b></font> add to {{{5}}}



Now replace the middle term {{{5c}}} with {{{-3c+8c}}}. Remember, {{{-3}}} and {{{8}}} add to {{{5}}}. So this shows us that {{{-3c+8c=5c}}}.



{{{c^2+highlight(-3c+8c)-24}}} Replace the second term {{{5c}}} with {{{-3c+8c}}}.



{{{(c^2-3c)+(8c-24)}}} Group the terms into two pairs.



{{{c(c-3)+(8c-24)}}} Factor out the GCF {{{c}}} from the first group.



{{{c(c-3)+8(c-3)}}} Factor out {{{8}}} 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.



{{{(c+8)(c-3)}}} Combine like terms. Or factor out the common term {{{c-3}}}







So our expression goes from {{{2(c^2+5c-24)}}} and factors further to {{{2(c+8)(c-3)}}}




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Answer:


So {{{2c^2+10c-48}}} completely factors to {{{2(c+8)(c-3)}}}