Solve the equation $4\sqrt{2-x^2}=-x^3-x^2+3x+3$ Solve the equation in $\Bbb R$:
$$4\sqrt{2-x^2}=-x^3-x^2+3x+3$$
Is there a unique solution $x=1$? I have trouble when I try to prove it. 
I really appreciate if some one can help me. Thanks!

It can be seen that $4 \sqrt{2 - x^2} = (1+x)(3 - x^2)$ leads to the expanded form, after squaring both sides and combining terms,
$$ x^6 + 2 x^5 - 5 x^4 - 12 x^3 + 19 x^2 + 18 x - 23 = 0 .$$
It is readily identified that $x=1$ is a solution and can then be factored out leading to
$$ (x-1)( x^5 + 3 x^4 - 2 x^3 - 14 x^2 + 5x + 23) = 0.$$
The polynomial of order $5$ has one real solution and $4$ complex solutions. 
