Let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively, where $f(x) = xe^x$. For non-zero real parameters $a$ and $b$, we consider the equation with unknown $x \in \mathbb { R }$
$$\mathrm { e } ^ { a x } + b x = 0 \tag{I.3}$$
Determine, according to the values of $a$ and $b$, the number of solutions of (I.3). Explicitly express the possible solutions using the functions $V$ and $W$.