Finding coefficients in expansions

Question

Show that the coefficient of $x^{−12}$ in the expansion of
$$\left(x^4−\frac{1}{x^2}\right)^5\left(x−\frac{1}{x}\right)^6$$ is $−15$, and calculate the coefficient of $x^2$. Hence, or otherwise, calculate the coefficients of $x^4$ and $x^{38}$ in the expansion of $$(x^2−1)^{11}(x^4+x^2+1)^5.$$

The first part of this exercise is solved easily considering "partitions", i.e. what can be added to get the exponents and turns out to be $-15$ and $215$ respectively. I was able to solve the second part of this exercise easily using multinomial theorem. However, this exercise is given before multinomial theorem is introduced, and I would really like to know what the "Hence" was gunning for. Once I factor out to get$$x^{21}\left(x-\frac{1}{x}\right)^{11}\left(x^2+\frac{1}{x^2}+1\right)^5,$$I can see similarities with the first question. I can even deduce that since $4-21=-17$, $38-21=17$, and inspecting signs, that the coefficients are equal in magnitude and opposite in sign. Please explain how I can use the first part of the question to proceed, rather than going "otherwise"/multinomial theorem.

Answer 1

\begin{align*} (x^2 - 1)^{11}(x^4 + x^2 + 1)^5 &=(x^2 - 1)^{5}(x^4 + x^2 + 1)^5(x^2 - 1)^{6} \\ &=(x^6 - 1)^5(x^2 - 1)^6 \\ &=x^{16}\left(x^4 - \frac{1}{x^2}\right)^5\left(x - \frac{1}{x}\right)^6 \end{align*} Thus, the coefficient of $x^4$ in the new equation is the same as the coefficient of $x^{-12}$ in the original equation, which is $-15$.

And like you mentioned, the coefficient of $x^4$ and $x^{42 - 4} = x^{38}$ should be the same but opposite in sign. So the coefficient of $x^{38}$ is $15$.

Answer 2

In $$(x^4-\frac{1}{x^2})^5(x-\frac{1}{x})^6=\frac{(x^6-1)^5(x^2-1)^6}{x^{16}}=\frac{(y^3-1)^5(y-1)^6}{y^8} $$ notice that every $x$ is in power of 2, so let $y=x^2$ then create the polynomial $$ P(y)=(y^3-1)^5(y-1)^6=(y^2+y+1)^5(y-11)^{11} $$ to which you try to find coefficients to $y^2=x^4$ and $y^9=x^{18}$. It was too tedious to expand by hand, so using the expansion calculator https://www.dcode.fr/math-expression-expand I got quite large numbers.