$\newcommand{\la}{\lambda}\newcommand{\R}{\mathbb{R}}$Let $I:=(0,1)$ and $Q=I^n$. Let $\la$ denote the Lebesgue measure over $\R^n$. 
Let $P_j$ denote the partial derivative of $P$ wrt to its $j$th argument. 
Because the polynomial $P=P(x_1,\dots,x_n)$ is non-constant, at least one of the $P_j$'s is a nonzero polynomial. Without loss of generality, $P_1$ is a nonzero polynomial. 

It follows by the main theorem of algebra and the Tonelli theorem that the Lebesgue measure $\la(N)$ of the (closed in $Q$) set 
$$N:=\{(x_1,\dots,x_n)\in Q\colon P_1(x_1,\dots,x_n)=0\}$$
of the zeros of the nonzero polynomial $P_1$ is $0$. 

Consider now the transformation 
$$Q\ni(x_1,\dots,x_n)\mapsto g(x_1,\dots,x_n):=(P(x_1,\dots,x_n),x_2,\dots,x_n)\in\R^n, $$
the sets
\begin{equation}
	Z_{(y_1,\dots,y_n)}:=\{x_1\in I\colon(x_1,y_2,\dots,y_n)\notin N, P(x_1,y_2,\dots,y_n)=y_1\}
\end{equation}
for $(y_1,\dots,y_n)\in\R^n$, and the sets  
\begin{equation}
	A_m:=\{(y_1,\dots,y_n)\in\R^n\colon\text{card}\,Z_{(y_1,\dots,y_n)}=m\}
\end{equation}
for $m=0,\dots,d$, where $\text{card}$ denotes the cardinality and $d$ is the degree of $P$ wrt $x_1$. Note that $(A_0,\dots,A_d)$ is a partition of $\R^n$, and the sets $A_1,\dots,A_m$ are open. 

By the inverse function theorem, for each $m\in\{0,\dots,d\}$, each $(y_1,\dots,y_n)\in A_m$, and each integer $k\in[1,m]$, there is a real-analytic function $X_{m,k}$ defined on a neighborhood $U_{(y_1,\dots,y_n)}$ of the point $(y_1,\dots,y_n)\in A_m$ such that 
\begin{equation}
	g(X_{m,k}(z_1,\dots,z_n),z_2,\dots,z_n)=(z_1,\dots,z_n)
\end{equation}
for all $(z_1,\dots,z_n)\in U_{(y_1,\dots,y_n)}$, and the value of the Jacobian determinant of the map
\begin{equation}
U_{(y_1,\dots,y_n)}\ni(z_1,\dots,z_n)\mapsto(X_{m,k}(z_1,\dots,z_n),z_2,\dots,z_n)
\end{equation}
at the point $(z_1,\dots,z_n)$ is 
\begin{equation}
	\partial_{z_1}X_{m,k}(z_1,\dots,z_n)=\frac1{P_1(X_{m,k}(z_1,\dots,z_n),z_2,\dots,z_n)}. 
\end{equation}

So, by the change of variables in multiple integrals, the density (say $f$) of the pushforward $g_\#(\la\downharpoonright Q)$ under $g$ of the measure $\la\downharpoonright Q$ wrt to $\la$ is given by the formula 
\begin{equation}
	f(y_1,\dots,y_n)=\sum_{m=1}^d 1((y_1,\dots,y_n)\in A_m)
	\sum_{k=1}^m\frac1{|P_1(X_{m,k}(y_1,\dots,y_n),y_2,\dots,y_n)|}
\end{equation}
for $(y_1,\dots,y_n)\in\R^n\setminus g(N)$. Note that $\la(g(N))=0$, since $\la(N)=0$ and $g$ is Lipschitz. So, we can let $f:=0$ on $g(N)$. 
 
Now, the density (say $f_1$) of the pushforward $P_\#(\la\downharpoonright Q)$ of the measure $\la\downharpoonright Q$ under $P$ wrt to $\la$ is given by the formula 
\begin{equation}
	f_1(y_1)=\int_{\R^{n-1}}dy_2\,\cdots\, dy_n\,f(y_1,\dots,y_n)
\end{equation}
for $y_1\in\R$.