Let's say I am doing the following simplification. $$x[n]= \delta[n] - \alpha\delta[n-1] $$ $$h[n] = \alpha^nu[n]$$ $$x[n]*h[n] = (\delta[n] - \alpha\delta[n-1])* \alpha^nu[n]$$ $$= (\alpha^n\delta[n] - \alpha^{n+1}\delta[n-1])*u[n]$$ $$=(\alpha^nu[n] - \alpha^{n+1}u[n-1])$$
However, I'm unable to simplify it further. But the end result should be purely $\delta[n]$. Where am I going wrong? I have a feeling it might be how I am carrying over $\alpha$; however, I fail to see how that just cancels out to purely the delta function.
Just use the fact that
$$f[n]\star\delta[n-k]=f[n-k]\tag{1}$$
for any sequence $f[n]$ (where $\star$ denotes discrete-time convolution). Consequently,
$$\begin{align}h[n]\star x[n]&=h[n]-\alpha h[n-1]\\&=\alpha^nu[n]-\alpha\alpha^{n-1}u[n-1]\\&=\alpha^n(u[n]-u[n-1])\\&=\alpha^n\delta[n]\\&=\delta[n]\end{align}$$