Observe?

Suppose my original non-negative integer is $x$ . The expression becomes

$4(x+1) +x^2$

Which is a perfect square.

From the given options, 4,441,400 are perfect squares. But 401 is not a perfect square.

Hence 401 could not be my new latest number.

The whole expression is equivalent to: $4(x+1)+x^2=4x+4+x^2$ Then we have to check out the option which doesn't give an integral solution for $x$ . Let us try $401$ : $x^2+4x+4=401$ $\Rightarrow\space x=\sqrt{401}-2\space \text{or}\space x=-2-\sqrt{401}$ $\text{And we are done!!}$