Answer
$$x=1,-3$$
Work Step by Step
$$\sqrt{-2x+3}=3-\sqrt{x+3}$$
$$-2x+3=9-6\sqrt{x+3}+(\sqrt{x+3})^2$$
$$-2x+3=12-6\sqrt{x+3}+x$$
$$-3x-9=-6\sqrt{x+3}$$
$$x+3=2\sqrt{x+3}$$
Square both sides.
$$x^2+6x+9=4(x+3)$$
$$x^2+6x+9=4x+12$$
$$x^2+2x-3=0$$
$$(x+3)(x-1)=0$$
$$x=1,-3$$