Short Answer x = - 6
Remark
You gave the second best answer.
The trick here is not to divide both sides by 2. Solve the problem this way.
log_5(x + 1)^2 = 2 Take the antilog of both sides
(x + 1)^2 = 5^2 Expand the equation
x^2 + 2x + 1 = 25 Subtract 25 from both sides.
x^2 + 2x - 24 = 0 Factor
(x + 6)(x - 4) = 0 Find the zeros.
x + 6 = 0
x = - 6 <<<<<<<< Answer. This is the extraneous root.
The reason this is an extraneous root is that x<=0 do not have a logarithem
x - 4 = 0
x = 4 This is a legitimate result to the original equation.
