A nonconstant polynomial $q$ with $q(0)>1$ attains infinitely many composite values at integers Let $q(n)$ be a nonconstant polynomial with integer coefficients, and let $c=q(0)$ be the constant term of $q$. Show that if $q$ is nonconstant and $c \gt 1$, then there are infinitely many $q(n)$ $\in \mathbb N$ that are not primes.
Hint: You may assume the familiar fact that the magnitude of any non constant polynomial, $q(n)$, grows unboundedly as $n$ grows. 

How to solve it using the given hint? As it is stated that polynomial has integer coefficients, so that means polynomial can be decreasing as well. But in that case we may not get mapping to infinitely many natural numbers.
 A: First, please note that some authors allow for negative numbers to be prime. Maybe your problem allows for that.
Actually, the thing that matters the most in the end is the behavior of the leading coefficient of the polynomial, say, $a_nx^n$ ; as $\mathbb N \rightarrow \infty$ , the poly will go either to  $(+/-) \infty $. Since your coefficients are natural numbers, your polynomial will go to $+\infty$ as you approach $+ \infty $. This means that, in the process, it will take-on infinitely-many values. Then, let $c:=q(0)$. Then $f(n.c)$, for n in $\mathbb N$ will be a multiple of c, i.e., $f(n.c)$ will never be prime . 
A: Suppose that $q(n)$ is prime for some $n$. Notice that $q(x)=a_mx^m+a_{m-1}x^{m-1}+\ldots+a_1x+c$. What can you say about $q(cn)$?
Your result will hold for any $c$. Split into cases of $c$ being an integer and noninteger. There is also the special case of $c=\pm 1$.
A: If there is no constant coefficient, then $n\mid q(n)$ for all $n$. If $q(n)$ is exactly $n$, then I claim that you are done (do you see why?). If not, then In general $q(n)$ will differ fom $n$ infinitely often, and you are done again.
If there is a constant coefficient $c$, then $q(nc)$ will be divisible by $c$ always.
In both cases, since polynomials take on arbitrarily large values, the above argument shows that $q(n)$ will take on infinitely many non-prime values.
