Question 31
Let $n$ be a positive integer. Determine all positive real numbers $x$ for which
$$
\frac{2^2}{x+1} + \frac{3^2}{x+2} + \dots + \frac{(n+1)^2}{x+n} + n x^2 = n x + \frac{n(n+3)}{2}
$$
$$
\frac{2^2}{x+1} + \frac{3^2}{x+2} + \dots + \frac{(n+1)^2}{x+n} + n x^2 = n x + \frac{n(n+3)}{2}
$$