How would one solve the following equation? This equation is giving me a hard time.
$$e^x(x^2+2x+1)=2$$
Can you show me how to solve this problem algebraically or exactly? I managed to solve it using my calculator with one of its graph functions. But I would like to know how one would solve this without using the calculator.
Highly appreciated,
Bowser.

$$
e^x(x+1)^2=2\implies e^{(x+1)/2}(x+1)/2=\sqrt{e/2}\implies(x+1)/2=\mathrm{W}\!\left(\sqrt{e/2}\right)
$$
Therefore,
$$
x=2\mathrm{W}\!\left(\sqrt{e/2}\right)-1
$$
where $\mathrm{W}$ is the Lambert W function.
There is an iterative algorithm given in this answer to compute $\mathrm{W}$.
Alternatively, N[2LambertW[Sqrt[E/2]]-1,20] in Mathematica yields 
$$
x=0.24879269668640244047
$$
