$n^4+2n^3+2n^2+2n+1 = (n^2+1)(n+1)^2$ If $n \geq 1$ composite, if $n \leq -3$ composite, small cases $n = 0$ gives $1$ not prime, $n = -1$ gives $0$ not prime, $n = -2$ gives $5$ which is prime. So there is one such integer.