  We consider the following class of focusing $L^2$-supercritical fourth-order
nonlinear Schr\"odinger equations
  \[
  i\partial_t u - \Delta^2 u + \mu \Delta u = - |u|^\alpha u, \quad (t,x) \in
\mathbb{R} \times \mathbb{R}^N,
  \]
  where $N\geq 2$, $\mu \geq 0$, and $\frac{8}{N}<\alpha<\alpha^*$ with
$\alpha^*:=\frac{8}{N-4}$ if $N\geq 5$ and $\alpha^*=\infty$ if $N\leq 4$. By
using the localized Morawetz estimates and radial Sobolev embedding, we
establish the energy scattering below the ground state threshold for the
equation with radially symmetric initial data. We also address the existence of
finite time blow-up radial solutions to the equation. In particular, we show a
sharp threshold for scattering and blow-up for the equation with radial data.
Our scattering result not only extends the one proved by Guo \cite{Guo}, where
the scattering was proven for $\mu = 0$, but also provides an alternative
simple proof that completely avoids the use of the concentration/compactness
and rigidity argument. In the case $\mu > 0$, our blow-up result extends an
earlier result proved by Boulenger-Lenzmann \cite{BL}, where the finite time
blow-up was shown for initial data with negative energy.
