Since $x$ is the product of four consecutive integers, it is always divisible by 4, i,.e., it is always even. So, $1 + x$ is always odd.
\Rightarrow 1+x& =y^4+2y^3-y^2-2y+1 \\
So, $1 + x$ is a perfect square as we can see. Hence, option A is the correct choice.
Note: In an exam, even if you are not able to work mathematically as above, you should not leave out this type of question. You should take two or three numerical values for $x$ and check which of the choices will be satisfiÂed. Take the four consecutive integers as $(1, 2, 3, 4)$, $(2, 3, 4, 5)$ and $(3, 4, 5, 6)$ and in each case, we find that $1+ x$ is a perfect square and odd. Then, we can mark (A) as the answer choice.