Generalization Newton's Binomial

Question

I'm trying to generalize Newton's binomial.
I got this result. Can I prove this by induction?

$$\prod_{k=1}^n (a_k+b_k) = \prod_{c=1}^n a_c+ \sum_{i=1}^{n-1}\sum_{d=1}^{\binom{n}{i}}\prod_{e=d}^{n-i} a_{e}\prod_{j\not=e}^{}b_{j}+\prod_{l=1}^n b_{l}.$$ $$(a + b)^n = a^n + \sum_{i=1}^{n-1}\binom{n}{i}a^{n-i}b^i + b^n$$

Thanks!

Answer 1

If $a_k = a$ and $b_k = b$ for every $k$ and $n=2$, then the LHS becomes $(a+b)^2$ and RHS $$ a^2 + \sum_{i=1}^{2-1}\sum_{d=1}^{2\choose i}a^{2-i-d}b^{i-1} + b^2 = a^2 + a^0 + a^{-1} + b^2. $$