I don't know if this is a good way to do this, but...
First, you can factor the numerator, which is \(15x^2-8x-7\), to \(\left(x-1\right)\left(15x+7\right)\).
We can do the same thing with the denominator \(\:x^2-2x+1\), to \(\left(x-1\right)^2.\)
Now, we have \(\frac{\left(x-1\right)\left(15x+7\right)}{\left(x-1\right)^2}\). And, we can cancel \(x-1\), since that is our common factor. Thus, we are left with \(\boxed{\frac{15x+7}{x-1}}\).
If you need more explanation on the factoring, just tell me. Also, I think synthetic division works.