52 Given: $n$ is a given positive integer, $x$ is a real number. Try to calculate: $\sum_{k=0}^{n} k^{2} \cdot C_{n}^{k} \cdot x^{k} \cdot(1-x)^{n-k}$. (Please write the answer in its simplest form.)


$52 \begin{aligned} & \sum_{k=0}^{n} k^{2} \cdot C_{n}^{k} \cdot x^{k} \cdot(1-x)^{n-k} \\ = & \sum_{k=1}^{n} k \cdot n \cdot C_{n-1}^{k-1} \cdot x^{k} \cdot(1-x)^{n-k} \\ = & n \cdot x \cdot \sum_{k=1}^{n} k \cdot C_{n-1}^{k-1} \cdot x^{k-1} \cdot(1-x)^{n-k} \\ = & n x \cdot\left(\sum_{k=1}^{n}(k-1) \cdot C_{n-1}^{k-1} \cdot x^{k-1} \cdot(1-x)^{n-k}+\sum_{k=1}^{n} C_{n-1}^{k-1} \cdot x^{k-1} \cdot(1-x)^{n-k}\right) \\ = & n x \cdot\left(\sum_{k=0}^{n-1} k \cdot C_{n-1}^{k} \cdot x^{k} \cdot(1-x)^{n-1-k}+\sum_{k=1}^{n-1} C_{n-1}^{k} \cdot x^{k} \cdot(1-x)^{n-1-k}\right) \\ = & n x \cdot\left((n-1) \cdot \sum_{k=1}^{n-1} C_{n-2}^{k-1} \cdot x^{k} \cdot(1-x)^{n-1-k}+(1-x+x)^{n-1}\right) \\ = & n x \cdot\left((n-1) \cdot x \cdot \sum_{k=0}^{n-2} C_{n-2}^{k} \cdot x^{k} \cdot(1-x)^{n-2-k}+1\right) \\ = & n x \cdot\left((n-1) x \cdot(1-x+x)^{n-2}+1\right) \\ = & n x \cdot(1+(n-1) x) \\ = & n \cdot(n-1) x^{2}+n \cdot x .\end{aligned}$