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