Let $k$ be an integer such that $2n^2 = dk$. Then we have
$$\begin{aligned} k^2(n^2 + d) &= k^2n^2 + k^2d = k^2n^2 + 2kn^2 \\ &= n^2(k^2 + 2k) = n^2((k+1)^2 - 1). \end{aligned}$$
Expression $(k+1)^2 - 1$ is not a square of an integer for $k \in \mathbb{N}$. Then, $k^2(n^2 + d)$ is not a square of an integer, so $n^2 + d$ cannot be a square of an integer.