Find exact roots of $\sqrt{4+x}+\sqrt{1-x}=x^2+3$

Question

I have some difficulties with this problem:

Find exact roots of $\sqrt{4+x}+\sqrt{1-x}=x^2+3$

I know that we can square both side 2 times to have 8th degree equation with x, but I can't find the excat form of roots. Is there another way to solve this problem (without square 2 times) or find the exact roots? Thank you so much

Answer 1

The Galois group of $x^7+12x^5+44x^3+52x+12$ is $S_7$, which is not solvable.

Edit: Possibly the question is

Find exact roots of $\sqrt{4+x}+\sqrt{1-x}=\sqrt{x^2+3}$

This is equivalent to finding the roots of $x^4+12x-12$, which has solvable Galois group $S_4$.