Task B-1.1. Factorize the expression

$$
a^{5}+b^{5}-a b\left(a^{3}+b^{3}\right)
$$
Solution. Let's rewrite the expression in another way,

$$
a^{5}+b^{5}-a b\left(a^{3}+b^{3}\right)=a^{5}+b^{5}-a^{4} b-a b^{4}
$$

group the terms and factor out the common factor

$$
a^{4}(a-b)-b^{4}(a-b)=(a-b)\left(a^{4}-b^{4}\right)
$$

Further factorization yields

$$
=(a-b)\left(a^{2}-b^{2}\right)\left(a^{2}+b^{2}\right)=(a-b)^{2}(a+b)\left(a^{2}+b^{2}\right)
$$