Add 1 to both sides:
[tex] \sqrt{x+3} = x+1 [/tex]
In cases like this, we have to remember that a root is always positive, so we can square both sides only assuming that
[tex] x+1 \geq 0 \iff x \geq -1 [/tex]
Under this assumption, we square both sides and we have
[tex] x+3 = (x+1)^2 \iff x+3 = x^2+2x+1 \iff x^2+x-2 = 0 [/tex]
The solutions to this equation are
[tex] x = -2,\ x=1 [/tex]
But since we can only accept solutions greater than -1, we discard [tex] x=-2[/tex] and accept [tex] x=1 [/tex].
In fact, we have
[tex] x=-2 \implies \sqrt{-2+3}-1=0\neq -2 [/tex]
and
[tex] x=1 \implies \sqrt{4}-1=1 [/tex]
which is the only solution.
