<s>[INST] Formula for monic quadratic polynomials over $\mathbb{Z}$ Let $p(x)$ be a monic quadratic polynomial over $\mathbb{Z}$. Show that,for any integer $n$, there exists an integer $k$ such that
     $(p(n))(p(n+1))=p(k)$
 [/INST] Let P(x)= x^2 + bx + c
Therefore, P(n)= n^2 + bn + c, for an arbitrary integer n
P(n+1) = (n+1)^2 + b(n+1) + c
=  n^2 + 2n + 1 + bn + b + c
= P(n) + 2n + b + 1
Therefore, P(n)P(n+1) = P(n)*{P(n) + 2n + b + 1}
= {P(n)}^2 + 2nP(n) + bP(n) + P(n)
= {P(n)}^2 + 2nP(n) + n^2 + bn + bP(n) +c
= (P(n) + n}^2 + b{P(n) + n} + c
= k^2 + bk + c , where k = P(n) + n,  (i.e) an integer,
= P(k)
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