Equations such as $\sqrt{x+1}+\sqrt{x+2}=x+3$ are easily solvable by squaring both sides. But if we increase an extra square root, like if trying to solve $\sqrt{x+1}+\sqrt{x+2}+\sqrt{x+3}=x+4$ we encounter a problem. Squaring both sides doesn't work now, since the L.H.S. would still end up with three terms involving square roots, rather than only one as for the previous equation.

That is, unless I'm missing something. Is there a way of solving equations such as the following analytically?

$\sqrt{P_1(x)}+\sqrt{P_2(x)}+\cdots+\sqrt{P_n(x)}=Q(x)$ where $n>2$ and $Q(x), P_1(x), \ldots, P_n(x)$ are all polynomials.