<s>[INST] Factoring $(x+1)(x+2)(x+3)(x+4)+1$ I have to factor this polynomial:
$$(x+1)(x+2)(x+3)(x+4)+1$$
WolframAlpha gives 
$$(x^2+5x+5)^2$$
How can I prove it without expanding the result?
Thanks!
 [/INST] I answered a similar question here. You can do it like this:
$(x+2)(x+3) = x^2+5x+5+1$, $(x+1)(x+4) = x^2+5x+5-1$, so
$$
(x+1)(x+2)(x+3)(x+4) = (x^2+5x+5+1)(x^2+5x+5-1) = (x^2+5x+5)^2-1^2.
$$
This gives the answer $(x+1)(x+2)(x+3)(x+4)+1 =(x^2+5x+5)^2 $.
</s>