Let $y_i = \frac{1}{x_0 - b_i} > 0$ for $i=1,\cdots,n$. Then \[P(x_0 + 1) = \prod_{i=1}^n (x_0-b_i +1) = \prod_{i=1}^n \left(\frac{1}{y_i} + 1 \right) = \frac{(y_1 +1) \cdots (y_n +1)}{y_1 \cdots y_n}\]Since $y_i + 1 = \underbrace{\frac{y_i}{n-1} + \cdots + \frac{y_i}{n-1}}_{(n-1)\text{-times}} + 1 \underset{\text{AM-GM}}{\geq} \frac{n}{(n-1)^{(n-1)/n}} y_i^{(n-1)/n}$ and $y_1+\cdots + y_n \geq n (y_1 \cdots y_n)^{1/n}$, we have \[(\text{LHS}) \geq \frac{n^n}{(n-1)^{n-1}} \cdot n = \left(1 + \frac{1}{n-1} \right)^{n-1} \cdot n^2 \underset{\text{Bernoulli}}{\geq} 2n^2\]