Answer
$x=\frac{-3-\sqrt{17}}{2}$
Work Step by Step
$\sqrt {3-x}+2=1-x$
$3-x=x^2+2x+1$
$x^2+2x+1=3-x$
$x^2+3x-2=0$
By quadratic formula:
$x_{1,\:2}=\frac{-3\pm \sqrt{3^2-4\cdot \:1\cdot \left(-2\right)}}{2\cdot \:1}$
$x_{1,\:2}=\frac{-3\pm \sqrt{17}}{2}$
Test: $x=\frac{-3+\sqrt{17}}{2}$
$\sqrt {3-x}+2=1-x$
$\sqrt {3-\frac{-3+\sqrt{17}}{2}}+2=1-\frac{-3+\sqrt{17}}{2}$
$\:3.56155\ne 0.43844 $
Test: $x=\frac{-3-\sqrt{17}}{2}$
$\sqrt {3-x}+2=1-x$
$\sqrt {3-\frac{-3-\sqrt{17}}{2}}+2=1-\frac{-3-\sqrt{17}}{2}$
$4.56155= \:4.56155 $