First make note of the domain of the left hand side.
• √(x + 1) is defined for x + 1 ≥ 0, or x ≥ -1.
• √(2 + √(x + 1)) is defined for 2 + √(x + 1) ≥ 0. If x ≥ -1, then this condition is satisfied.
• √(7 + √(2 + √(x + 1))) is defined for 7 + √(2 + √(x + 1)) ≥ 0. This condition is also satisified automatically if x ≥ -1.
Now solve the equation. Taking the square of both sides of
√(7 + √(2 + √(x + 1))) = 3
yields
7 + √(2 + √(x + 1)) = 3² = 9
√(2 + √(x + 1)) = 2
Squaring both sides again gives
2 + √(x + 1) = 2² = 4
√(x + 1) = 2
And once more,
x + 1 = 2² = 4 ⇒ x = 3