yes, josh's solution is correct.
this is josh’s solution for the equation x^2 + 12x + 32 = 0:
x^2 + 12x +32 = 0
x^2 + 12x = -32 (32 is move to the other side that is why it became negative -32 and that is correct)
x^2 + 12x +36 = -32 + 36 (both sides are added with 36 so that the left side could be a perfect square: the process is completing the square: adding any the same number for both sides will result to the same original equation)
〖(x+6)〗^2 = 4 (the left side is a perfect square so it is right)
x + 6 = ±4 (square of any value is ± so it is right)
x=-2 (get the possible values of x)
Please understand the concept of an imaginary number line.