<s>[INST] How to integrate $\frac{4x+4}{x^4+x^3+2x^2}$? Please could anyone help me to integrate
$\quad\displaystyle{4x + 4 \over x^4 + x^3 + 2x^2}.\quad$
I know how to use partial fraction and I did this: 
$$
x^{4} + x^{3} + 2x^{2} = x^{2}\left(x^{2} + x + 2\right)
$$
And then ?.$\quad$ Thanks all.
 [/INST] Your idea is good, write
$$\frac{4x+4}{x^4+x^3+2x^2}=4\frac{x+1}{x^2 (x^2+x+2)}$$
Then, partial fraction will be of the form
$$\frac{x+1}{x^2 (x^2+x+2)}=\frac{a}{x} + \frac{b}{x^2} + \frac{cx+d}{x^2+x+2}$$
There are several methods to identify coefficients from this. The simplest IMHO is to rewrite the right hand side with the same denominator:
$$\frac{ax+b}{x^2} + \frac{cx+d}{x^2+x+2} = \frac{(ax+b)(x^2+x+2) + (cx+d)x^2}{x^2(x^2+x+2)}$$
Then develop, and write that the numerator must be $x+1$, thus you get equations by writing that the coefficients of $x^3$ and $x^2$ are $0$, and the coefficient of $x$ and the constant are $1$. Four linear equations, fours unknowns.
Is it enough for you to complete the process?
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