We know that his original age (34 years ago) + 56 years = the square of his original age. Let's call his original age X.
X + 56 = X^2
Now get all the Xs on one side:
56 = X^2 - X
We can simplify the right side of that like this:
56 = X (X - 1)
Now step back and think about it. What number can you multiply by the number that's 1 below it and end up with 56?
You can probably guess the answer, but if not, find the factors of 56:
1 x 56, 2 x 28, 4 x 14, 7 x 8
Clearly, 7 and 8 are the pair we need. So, X must be 8 (since X-1 has to be 7).
REMEMBER: The question asks for his present age.
8 + 34 = 42