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 \(x^2\).

Could we expand each of the brackets?

What contribution do we need from each of the bracketed terms to get \(x^{-12}\) or \(x^2\) terms?

Hence, or otherwise, calculate the coefficients of \(x^4\) and \(x^{38}\) in the expansion of \[(x^2 − 1)^{11}(x^4 + x^2 + 1)^5.\]

How are the two parts related? Could we rewrite this new expression to make it look like the first part?

What happens if we mulitply \((x^4+x^2+1)\) by \((x^2-1)\)?