Question 1059605
x(x+1)(x+2)(x+3)+1=x(x+3)(x+2)(x+1)+1
so (x^2+3x)(x^2+3x+2)+1=(x^2+3x+1)(x^2+3x+2)-(x^2+3x+2)+1
You are converting the first term to (x^2+3x+1), and that is adding x^2+3x+2 to that side.  That means you have to subtract it.
Convert the (x^2+3x+2) to (x^2+3x+1). But that requires adding the factor x^2+3x+1. because by subtracting one from that term, which is multiplied by (x^2+3x+1), you have to add it back.
Then you have (x^2+3x+1)(x^2+3x+1)-(x^2+3x+2)+(x^2+3x+1)+1.  The last two terms disappear because distributing the minus sign give -x^2-3x-2+x^2+3x+1+1.
You are left with (x^2+3x+1)^2, which is a perfect square.
Pick an x, like 7
7,8,9,10 has a product of 5040 and 5041, 1 more, is a perfect square of 
x^2+3x+1, or 49+21+1, or 71.