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 $$ (x^2-1)^{11}(x^4+x^2+1)^5. $$
