Probability density function

$f(x)=(2pi)^{-n/2}$$e^{frac{-sum_{i=1}^nx_i^{2}}{2}}$ $times$ ($1$ + $Pi_{i=1}^n$$x_i$$e^{frac{-x_i^{2}}{2}}$)

$Xin$ $mathbb{R^{n}}$ Is this multivariate normal?
Does $(X_1, X_2,…, X_n)^{T}$ have a multivariate normal distribution?