If x is a perfect square (i.e., the square of some integer), then the next perfect square larger than x is:
- x + 1
- x2 + 1
- x2 + 2x + 1
- x2 + x
- x + 2√x + 1
Let x = n2. Then √x = n. The next perfect square after x will be the square of n + 1. So we want (n + 1)2 in terms of x.
(n + 1)2 = n2 + 2n + 1 = x + 2√x + 1. The answer is (e).
Check: Let x = 49; then n = 7. So n + 1 = 8, and (n + 1)2 = 64.
Is 64 the next perfect square after 49? Yes.
Does x + 2√x + 1 = 64? Yes, 49 + 2 · 7 + 1 = 49 + 14 + 1 = 64.