Rearranging terms for $x$ involving $e^x e^y = (1 + e^x)^2$

Question

How can we solve this equation for $x$?
$$4e^x e^y = ( 1 + e^x)^2$$

I got stuck with this problem.
It took a lot of time, however, each time unable to solve.

Attempt:
put e^x = u i.e, x = ln(u) $$4u e^y = ( 1 + u)^2$$ $$4u e^y = 1 + 2u + u^2$$ $$0 = u^2 + (2-4e^y).u + 1$$ $$ 2u = -(2-4e^y) +- \quad \sqrt{(2-4e^y)^2 - 4} $$ $$ 2e^x = -(2-4e^y) +- \quad \sqrt{(2-4e^y)^2 - 4} $$ $$ ln2 + x = ln[-(2-4e^y) +- \quad \sqrt{(2-4e^y)^2 - 4}] $$

Then again it stuck!

Any help will be truly appreciated!!

Answer 1

Hint: Let $u=e^x$, then we get a quadratic equation $4ue^y=(1+u)^2$ which you can solve with the quadratic formula. Then back substitute for $x$.