I have the following expression
$ 2x+3x^2+e^{5x+x^2}=7 $
which I need rearranged in a form of the type $Ye^Y=Z$ with Y a function of x and Z some constant. I have tried the substitution $y=5x+x^2$, that is I replaced x by the roots of that equation. For the root $x=\frac{-5+\sqrt{25+4y}}{2}$ I get
$e^y+11\sqrt{25+4y}+4y+38=0$
and I don't see how to get to the above mention form from here. Any idea about how I should tackle this? I don't know if the Taylor expansion of the exponential could help, can you see any link.Thank you.
Let $Z=7$ and consider $y=W(2x+3x^2+e^{5x+x^2})$ where $W$ is a Lambert Omega function. In general you have $y=W(2x+3x^2+e^{5x+x^2}-7+Z)$ where $z$ is just a chosen number. I don't see any simplification after this point.