HINT : Let $f(x)$ be an $n$- degree polynomial with integer coefficients.

Then $f(0)$ is the constant term and $f(1)$ is the sum of all coefficients and by the condition both are odd.

Now Lets assume $f(x)$ has a integer solution $x=c$

Two cases may arise, $c$ is either odd or even.

Try to prove that in both cases $f(c)$ is odd

and then our proof is done (How?? Think !!!)