<s>[INST] Find the remainder when $x^{10}+1$ is divided by $(x^2+1)(x^2+x+1)$ Find the remainder when $x^{10}+1$ is divided by $(x^2+1)(x^2+x+1)$
I have done it until the the divisor is of second degree. 
But here the degree of the remainder is $4$
This means the remainder will be of the form.     : $ax^3+bx^2+cx+d$ which makes it difficult to solve. Is it possible to solve $4$ variants with $4$ equations?
 [/INST] $$\dfrac{x^{10}+1}{(x^2-1)(x^2+x+1)}=(x-1)(x^5-x^3+x^2+x-1)+\dfrac{{x^3+x}}{(x^2-1)(x^2+x+1)}$$
</s>