Move square root term to one side:
[tex]2 \sqrt{x+4} = x+1[/tex]
Then square both sides:
[tex](2 \sqrt{x+4})^2 = (x+1)^2 \\ \\ 4(x+4) = x^2 +2x+1[/tex]
Move everything to one side so it equals 0. Combine like terms.
[tex]x^2 +2x+1 -4x-16 = 0 \\ \\ x^2 -2x -15 = 0[/tex]
Solve quadratic by factoring or using quadratic formula.
[tex]x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-15)}}{2} = \frac{2 \pm 8}{2} = -3,5 \\ \\ x^2 -2x-15 = (x-5)(x+3) =0[/tex]
Check for any extraneous solutions.
Plug answers back into original equation.
[tex]5+1-2\sqrt{5+4} = 6-2\sqrt{9} = 6-6 = 0 \\ \\ -3 +1 -2 \sqrt{-3+4} = -2-2\sqrt{1} = -4 \\ \\ x = 5, x \ne -3 [/tex]
