<s>[INST] Prove that the product of four consecutive positive integers plus one is a perfect square I need to prove the following, but I am not able to do it. This is not homework, nor something related to research, but rather something that came up in preparation for an exam.

If $n = 1 + m$, where $m$ is the product of four consecutive positive
  integers, prove that $n$ is a perfect square.

Now since $m = p(p+1)(p+2)(p+3)$;

$p = 0, n = 1$ - Perfect Square
$p = 1, n = 25$ - Perfect Square
$p = 2, n = 121$ - Perfect Square

Is there any way to prove the above without induction? My approach was to expand $m = p(p+1)(p+2)(p+3)$ into a 4th degree equation, and then try proving that $n = m + 1$ is a perfect square, but I wasn't able to do it. Any idea if it is possible?
 [/INST] Product of 4 consecutive numbers can be shown as
$$\begin{align}(x+1)(x+2)(x+3)(x+4)
    &=(x+1)(x+4)(x+2)(x+3) \\
    &=(x^2+5x+4)  (x^2+5x+6) \\
    &=\underset{A-B}{(x^2+5x+5 -1)} \underset{A+B}{(x^2+5x+5 +1)} \\
    &=\underset{A^2-B^2}{(x^2+5x+5)^2 - 1}\end{align}$$
Where $(x^2+5x+5)^2$ is a perfect square.
Proved.
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