Partial Fraction Decomposition of a Polynomial division

Question

Question :Write $$\frac{x^5}{(x^2+1 )(x+1)^2}$$ as a sum of partial fraction

What I've tried is to do polynomial long division twice to reduce the degree of numerator to be smaller than denominator than carry on to the normal steps of partial fraction decomposition

this is what i get $$x - 2 + \frac{1}{2x^2+2} + \frac{2}{x+1} - \frac{1}{2(x+1)^2}$$

but i'm not sure if its correct

Answer 1

$$x-2+\frac{1}{2(x^2+1)}+\frac{2}{x+1}-\frac{1}{2(x+1)^2}$$ $$=\frac{2(x-2)(x^2+1)(x+1)^2+(x+1)^2+4(x^2+1)(x+1)-(x^2+1)}{2(x^2+1)(x+1)^2}$$ $$=\frac{2x^5}{2(x^2+1)(x+1)^2}=\frac{x^5}{(x^2+1)(x+1)^2}$$ So, yours is correct.

Answer 2

Hint: $x^5 = x^3(x^2+1) - x^3$, $x^3 = (x^3+1) - 1 = (x+1)(x^2-x+1) - 1$. Then you "slowly" split the fraction to two fractions that can be simplified.