NZ-NZMO 2022 Problem 8

Find all real numbers $x$ such that $-1 < x \leq 2$ and

$$\sqrt{2 - x} + \sqrt{2 + 2x} = \sqrt{\frac{x^{4} + 1}{x^{2} + 1}} +\frac{x + 3}{x + 1}.$$