  Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ where $p$ is an
odd prime, $n$ be a positive integer satisfying ${\rm gcd}(n,p)=1$, and denote
$R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ where $e\geq 4$ be an even integer.
Let $\delta,\alpha\in \mathbb{F}_{p^m}^{\times}$. Then the class of
$(\delta+\alpha u^2)$-constacyclic codes over $R$ is a significant subclass of
constacyclic codes over $R$ of Type 2. For any integer $k\geq 1$, an explicit
representation and a complete description for all distinct $(\delta+\alpha
u^2)$-constacyclic codes over $R$ of length $np^k$ and their dual codes are
given. Moreover, formulas for the number of codewords in each code and the
number of all such codes are provided respectively. In particular, all distinct
$(\delta+\alpha u^2)$-contacyclic codes over $\mathbb{F}_{p^m}[u]/\langle
u^{e}\rangle$ of length $p^k$ and their dual codes are presented precisely.
