I'm trying to prove $$\sum_{j=1}^{n-1} jx^j = \frac{x-nx^n+(n-1)x^{n+1}}{1-x^2}$$ holds for all positive integers $n$ and real $x\ne \pm 1$ by induction.
In the inductive step I get
$$\begin{align}\sum_{j=1}^{k} jx^j &= \frac{x-kx^k+(k-1)x^{k+1}}{1-x^2} +kx^k \\ &= \frac{x-kx^k+(k-1)x^{k+1}+kx^k(1-x^2)}{1-x^2} \\ &= \frac{x+(\color{red}{k}-1)x^{k+1}\color{red}{-k}x^{k+2}}{1-x^2}\end{align}$$
Which is close to what I want, but for this to be true the two red $k$'s would have to be the opposite sign.
Have I made some mistake?
Your mistake is in the problem statement. You're trying to prove a false statement. Plug in $n=2$ to see why; the denominator is incorrect.
By the way, to do this without induction, note that the sum you seek is $$x \dfrac{d}{dx} \sum_{j=1}^{n-1} x^j = x \dfrac{d}{dx} \frac{x^n-x}{x-1}$$