Problem: Solve the equation:
$$x^2+23x+23=(x+2)\sqrt{2(x^2+3x+6)}$$
My attempt
After I squaring and simplifying, I got the following quartic equation:
$$x^4-32x^3-531x^2-986x-481=0$$
This equation has ugly roots, and it's way too hard to factor out. I've also tried to use Descartes' solution but failed. How do I solve the quartic equation? (Or is there another way to solve the original equation?)
Edit:
The above equation was wrong. This is the correct one:
$$4x^2+23x+23=(x+2)\sqrt{2(x^2+3x+6)}$$
This one seems better, I could just solve by doing the same as above (squaring and use Descartes' solution to solve the quartic equation). But is there a nicer way to solve the equation?
There's nothing you can do but solve the quartic numerically – there's no way to simplify the exact expressions. All the roots are real, so we check whether each root satisfies the original equation, and we find that three do: $$x=-10.392800\dots$$ $$x=-0.897551\dots$$ $$x=44.450403\dots$$