Can $n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$ be derived from the binomial theorem?

Start with

$$(1+x)^n = \sum_{i=0}^n \binom{n}{i} x^i$$

Take the derivative of both sides:

$$n (1+x)^{n-1} = \sum_{i=1}^n i \binom{n}{i} x^{i-1}$$

Multiply both sides by $x$:

$$n x (1+x)^{n-1}= \sum_{i=1}^n i \binom{n}{i} x^i$$

Take another derivative:

$$n (1+x)^{n-1} + n (n-1) x (1+x)^{n-2} = \sum_{i=1}^n i^2 \binom{n}{i} x^{i-1}$$

Plug in $x=1$ in the above equation:

$$n 2^{n-1} + n (n-1) 2^{n-2} = n (n+1) 2^{n-2} = \sum_{i=1}^n i^2 \binom{n}{i} $$

Hint: taking the first two consecutive derivatives

$$(1+x)^n=\sum_{k=0}^nx^k\binom{n}{k}\\n(1+x)^{n-1}=\sum_{k=1}^nkx^{k-1}\binom{n}{k}\\n(n-1)(1+x)^{n-2}=\sum_{k=2}^nk(k-1)x^{k-2}\binom{n}{k}\ldots$$

Start from the binomial theorem in the form

$$(x+1)^n=\sum_{k=0}^n\binom{n}kx^k$$

and differentiate with respect to $x$:

$$\begin{align*} n(x+1)^{n-1}&=\sum_{k=0}^n\binom{n}kkx^{k-1}\\\\ &=\sum_{k=1}^n\binom{n}kkx^{k-1}\;. \end{align*}$$

Differentiate again:

$$\begin{align*} n(n-1)(x+1)^{n-2}&=\sum_{k=1}^n\binom{n}kk(k-1)x^{k-2}\\\\ &=\sum_{k=1}^n\binom{n}kk^2x^{k-2}-\sum_{k=1}^n\binom{n}kkx^{k-2}\;. \end{align*}$$

Now let $x=1$ to get

$$\begin{align*} n(n-1)2^{n-2}&=\sum_{k=1}^n\binom{n}kk^2-\sum_{k=1}^n\binom{n}kk\\\\ &=\sum_{k=1}^n\binom{n}kk^2-\sum_{k=1}^n\binom{n-1}{k-1}n\\\\ &=\sum_{k=1}^n\binom{n}kk^2-n\sum_{k=0}^{n-1}\binom{n-1}k\\\\ &=\sum_{k=1}^n\binom{n}kk^2-n2^{n-1}\\\\ &=\sum_{k=1}^n\binom{n}kk^2-2n2^{n-2}\;, \end{align*}$$

and solve for $\displaystyle\sum_{k=1}^n\binom{n}kk^2$ to get the desired result.