Computing Discrete Convolution in terms of unit step function
As a first remark, I would really suggest to also familiarize yourself with the graphical way of convolving two functions or sequences. This will also make your calculations a lot easier because you get some intuition on the nature of the solution.
Second, your definitions of $x[n]$ and $h[n]$ in terms of the step sequence $u[n]$ are not correct. E.g., the given $x[n]$ equals $1$ for $n=8$, but $u[n-3]-u[n-8]$ is zero for $n=8$.
It is formally correct to represent the solution as the sum of four individual convolutions of shifted step sequences, so what you could do is derive a general solution of the individual convolutions of the form
$$u[n-n_0]\star u[n-n_1]=(n-N_0)u[n-N_1]\tag{1}$$
I leave it up to you to derive the correct values of $N_0$ and $N_1$ in terms of $n_0$ and $n_1$. Once you have obtained that result, you can just compute the solution to the original problem as the sum of four terms of the form $(1)$.