Wrong induction basis may give a completely wrong statement

Solution:

Part a

If $n^2+5n+1$ is even, then
\begin{equation}\label{1-11-1}
(n+1)^2+5(n+1)+1=(n^2+5n+1)+2n+6
\end{equation} is also even. Therefore $P_{n+1}$ is true whenever $P_n$ is true.

Part b

There is no integer $n$ such that $n^2+5n+1$ is even. It can be seen easily by looking at $P_1:1^2+5\cdot 1+1$ is even which is clearly wrong. In factor, using \eqref{1-11-1}, one can show that $n^2+5n+1$ is odd by induction. You just need to show it like Part a.