Finding the term in binomial theorem.

Question

Please, how do I solve for the term in $x^2$ and the term independent of $x$ in the expansion of $(x^2 - 2/x^2)^4$?

Answer 1

Hint. The general term (indexed by $k$) in the expansion of this expression is given by $$\binom{4}{k}(x^2)^{4-k}\left(\frac{-2}{x^2}\right)^k=\binom{4}{k}x^{8-2k}(-2)^kx^{-2k}=(-2)^k\binom{4}{k}x^{8-2k-2k}=(-2)^k\binom{4}{k}x^{8-4k}.$$

Set the exponent of $x$ equal to $2$ to find out the corresponding value of $k$ needed, and equal to $0$ to find the term independent of $x.$