4. $[\mathbf{1 0}]$ Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$such that
$$
f(x+f(y+x y))=(y+1) f(x+1)-1
$$
for all $x, y \in \mathbb{R}^{+}$.
$\left(\mathbb{R}^{+}\right.$denotes the set of positive real numbers.)