Can we find the coefficients of $x^{−12}$ and $x^2$ in this expansion?

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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 \[(x^2 − 1)^{11}(x^4 + x^2 + 1)^5.\]