How many integers n are there such that √n + (√n+4355) is rational?

Assuming that you mean √n + √(n + 4355):

First of all, we need n = x^2 for some integer x.

Next, we need n + 4355 = y^2 for some integer y

==> y^2 - x^2 = 4355

==> (y - x)(y + x) = 5 * 13 * 67.

Without loss of generality, we can take x > 0. Then, y - x < y + x.

Now, we simply consider all factorizations of 4355:

(i) y - x = 1 and y + x = 4355 ==> (x, y) = (2177, 2178)

(ii) y - x = 5 and y + x = 871 ==> (x, y) = (433, 438)

(iii) y - x = 13 and y + x = 335 ==> (x, y) = (161,174)

(iv) y - x = 65 and y + x = 67 ==> (x, y) = (1, 66).

Hence, there are four values of n (since n = x^2):

2177^2, 433^2, 161^2 and 1^2.

I hope this helps!