How to solve $x e^x + x = 1$?

I have seen a mathematics-related video here, which introduce a Lambert W function to the audience: $$f(x)=xe^x\\ W(x)=f^{-1}(x)$$

Then we can use $W(x)$ to solve some transcendental equation conveniently, such $x^x=4$. But he left homework at the end of the video:

To solve equation $x e^x + x = 1$

Well, this homework made me lose sleep for 2 days. I don't know how to solve this equation with the $W(x)$. Can someone please guide me?

The equation cannot be solved in terms of Lambert W.

To show this, try to rearrange the equation to a form $f(x)e^{f(x)}=c$, $f$ a function in the complex numbers and $c$ a complex constant.

$$xe^x+x=1$$

$$xe^x=1-x$$

$$\frac{x}{1-x}e^x=1$$

We see: the exponent is $x$, a polynomial in $x$, and the factor is not a polynomial in $x$. There is no chance to rearrange the equation by only elementary operations to a form where the factor is equal to the exponent. Therefore your equation is not in a form that can be solved by Lambert W.

Generalized Lambert W ([Mezö/Baricz 2017]) is needed to solve equations of your form in closed form.
$\ $

[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934

I don't believe this one can be solved using $W$. Maple does not find a closed-form solution, nor does Wolfram Alpha.