Pattern equation

We can write the equation in the form $\sum_{k = 2}^{2021} (x^2+kx+k-1)^2 = 0.$ Note that if $x_0 \in \mathbb{R}$ is such that $x_0^2+kx_0+k-1 = 0$ for every $k \in \{2, \ldots, 2021\}$ , then $x_0$ solves the original equation.

Observe, that for every $k \in \{2, \ldots, 2021\}$ the solutions of $x^2+kx+k-1 = 0$ are $x_1 = -1$ and $x_2 = -k+1$ (they are the same for $k = 2$ ).

Since, the only common solution of the subproblems is $-1$ , the answer is $-1$ .