Answer
$$\sum^{\infty}_{n=0} ((n+2)(n+1)a_{n+2} + na_n)x^n$$
Work Step by Step
1. Substitute each "m" in the first sum with a "m+2"
$$\sum^{\infty}_{m=0} (m+2)(m+1)a_{m+2}x^m$$
2. Write the second expression starting at k =0:
$$x \sum^{\infty}_{k=0}ka_kx^{k-1} = x\Big[(0)a_0x^{-1} \Big] + x \sum^{\infty}_{k=1}ka_kx^{k-1} $$
$$x \sum^{\infty}_{k=0}ka_kx^{k-1} = x \sum^{\infty}_{k=1}ka_kx^{k-1} $$
3. Write the second expression with the 'x' multiplying inside the sum:
$$ \sum^{\infty}_{k=0}ka_kx^{k-1}x= \sum^{\infty}_{k=0}ka_kx^{k}$$
4. Substitute:
$$\sum^{\infty}_{m=0} (m+2)(m+1)a_{m+2}x^m
+ \sum^{\infty}_{k=0}ka_kx^{k}$$
$$\sum^{\infty}_{n=0} (n+2)(n+1)a_{n+2}x^n + na_nx^n$$ $$\sum^{\infty}_{n=0} ((n+2)(n+1)a_{n+2} + na_n)x^n$$
