Solve Challenging Limit Homework: 2

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Homework Statement

Find this crazy limit using algebraic manipulations. I've tried quite a bit of stuff and keep getting lost, can you recommend anything?

The answer is 2.

Homework Equations

\lim_{x \to - 1} \frac{108(x^2 \ + \ 2x)(x \ + \ 1)^3}{(x^3 \ + \ 1)^3(x \ - \ 1)}

The Attempt at a Solution

\lim_{x \to - 1} \frac{108(x^2 \ + \ 2x)(x \ + \ 1)^2(x \ + \ 1)}{(x^3 \ + \ 1)^3(x \ - \ 1)}

\lim_{x \to - 1} \frac{108(x^2 \ + \ 2x)(x ^2\ + \ 2x \ + \ 1)(x \ + \ 1)}{(x^3 \ + \ 1)^3(x \ - \ 1)}

\lim_{x \to - 1} \frac{108(x^2 \ + \ 2x)(x ^3\ + \ 3x^2 \ + \ 3x \ + \ 1)}{(x^3 \ + \ 1)^3(x \ - \ 1)}

If I were to keep going expanding I'll get x^10 on the bottom and x^5 on the top, it just gets so hairy & I can't find any commonality.

 

[Gregg]

Simplify it to

\frac{108 x (2+x)}{(x-1) \left(1-x+x^2\right)^3}

 

[sponsoredwalk]

Ahh! I should have just realized to factor x³ + 1! So simple now thank you :)