_________ \/5+2х-х^2=х+1

$\sqrt{5+2x- x^{2} } = x-1 \\ (\sqrt{5+2x- x^{2} })^{2} = (x-1)^{2} \\ 5+2x- x^{2} = x^{2} - 2x + 1 \\ 2x^{2} - 4x - 4 = 0 \\ x^{2} - 2x - 2 = 0 \\ D = 4 + 8 = 12 \\ \sqrt{D} = 2 \sqrt{3} \\ x_{1} = \frac{2+ 2 \sqrt{3}}{2} = 1 + \sqrt{3}} \\ x_{2} = \frac{2- 2 \sqrt{3}}{2} = 1 - \sqrt{3}} \\$
Проверка корней:
$x=1+ \sqrt{3}} \\ \sqrt{5+2(1+ \sqrt{3} )- (1+ \sqrt{3}} )^{2} }} = (1+ \sqrt{3}} )-1 \\ \sqrt{5+2+ 2\sqrt{3}- 1- 2\sqrt{3}-3} }} = \sqrt{3}}\\ \sqrt{3}} = \sqrt{3}}\\$ (ВЕРНО)
$x=1- \sqrt{3}} \\ \sqrt{5+2(1- \sqrt{3} )- (1- \sqrt{3}} )^{2} }} = (1- \sqrt{3}} )-1 \\ \sqrt{5+2- 2\sqrt{3}- 1+ 2\sqrt{3}-3} }} = -\sqrt{3}}\\ \sqrt{3}} = - \sqrt{3}}\\$
(НЕВЕРНО)

Ответ: 1+√3.