Question 701292


{{{4x^2-56x-92}}} Start with the given expression.



{{{4(x^2-14x-23)}}} Factor out the GCF {{{4}}}.



Now let's try to factor the inner expression {{{x^2-14x-23}}}



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Looking at the expression {{{x^2-14x-23}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-14}}}, and the last term is {{{-23}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-23}}} to get {{{(1)(-23)=-23}}}.



Now the question is: what two whole numbers multiply to {{{-23}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-14}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-23}}} (the previous product).



Factors of {{{-23}}}:

1,23

-1,-23



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



These factors pair up and multiply to {{{-23}}}.

1*(-23) = -23
(-1)*(23) = -23


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-14}}}:



<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>-23</font></td><td  align="center"><font color=black>1+(-23)=-22</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>23</font></td><td  align="center"><font color=black>-1+23=22</font></td></tr></table>



From the table, we can see that there are no pairs of numbers which add to {{{-14}}}. So {{{x^2-14x-23}}} cannot be factored.



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Answer:



So {{{4x^2-56x-92}}} simply factors to {{{4(x^2-14x-23)}}}



In other words, {{{4x^2-56x-92=4(x^2-14x-23)}}}.