Rewrite the given expression as a sum whose generic term involves x^n:
$$ x\cdot\sum_{n=1}^{\infty}(n a_n x^{n-1}) + \sum_{k=0}^{\infty}(a_k x^{k} ) $$
I get a sum starting at one:
$$ \sum_{n=1}^{\infty}(n a_n x^{n}) + \sum_{n=1}^{\infty}(a_n x^{n} ) = \sum_{n=1}^{\infty}(n+1)a_n x^{n} $$
Whereas the answer is a similar sum starting at zero.
$$ \sum_{n=0}^{\infty}(n+1)a_n x^{n} $$
Where am I going wrong?
My advice is to write out the first several terms of the expression: $$x \sum_{n=1}^\infty na_n x^{n-1} + \sum_{k=0}^\infty a_k x^k = x(a_1 x^0 + 2a_2 x^1 + 3a_3 x^2 + \cdots) + (a_0 x^0 + a_1 x^1 + a_2 x^2 + \cdots).$$ Now equating like powers of $x$ gives the result $$a_0 x^0 + 2a_1 x^1 + 3a_2 x^2 + \cdots.$$ This suggests that the general term should have the form $(n+1)a_n x^n$, for $n = 0, 1, 2, \ldots$. All that remains is to formalize this by expressing the summands with corresponding indices of summation.