How to simplify this sequence
HINT:

The only thing you need to know is that any sequence convolved with a (shifted) delta impulse is a shifted version of itself:

$$f[n]*\delta[n-k]=f[n-k]$$

Apply this to the expression $u[n]*(\delta[n]-\delta[n-1])$, and realize that what you're left with is a very simple sequence. (The result of (a) will be helpful.)

Then convolve this very simple sequence with $x[n]$ to get the final result.