Question 117219

{{{x^3+6x^2+5x}}} Start with the given expression



{{{x(x^2+6x+5)}}} Factor out the GCF {{{x}}}



Now let's focus on the inner expression {{{x^2+6x+5}}}





Looking at {{{x^2+6x+5}}} we can see that the first term is {{{x^2}}} and the last term is {{{5}}} where the coefficients are 1 and 5 respectively.


Now multiply the first coefficient 1 and the last coefficient 5 to get 5. Now what two numbers multiply to 5 and add to the  middle coefficient 6? Let's list all of the factors of 5:




Factors of 5:

1,5


-1,-5 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 5

1*5

(-1)*(-5)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">5</td><td>1+5=6</td></tr><tr><td align="center">-1</td><td align="center">-5</td><td>-1+(-5)=-6</td></tr></table>



From this list we can see that 1 and 5 add up to 6 and multiply to 5



Now looking at the expression {{{x^2+6x+5}}}, replace {{{6x}}} with {{{1x+5x}}} (notice {{{1x+5x}}} adds up to {{{6x}}}. So it is equivalent to {{{6x}}})


{{{x^2+highlight(1x+5x)+5}}}



Now let's factor {{{x^2+1x+5x+5}}} by grouping:



{{{(x^2+1x)+(5x+5)}}} Group like terms



{{{x(x+1)+5(x+1)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{5}}} out of the second group



{{{(x+5)(x+1)}}} Since we have a common term of {{{x+1}}}, we can combine like terms


So {{{x^2+1x+5x+5}}} factors to {{{(x+5)(x+1)}}}



So this also means that {{{x^2+6x+5}}} factors to {{{(x+5)(x+1)}}} (since {{{x^2+6x+5}}} is equivalent to {{{x^2+1x+5x+5}}})


{{{x(x+5)(x+1)}}} Now reintroduce the GCF


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Answer:


So {{{x^2+6x+5}}} factors to {{{x(x+5)(x+1)}}}