<s>[INST] Finding real solutions to $\sqrt{x}+\sqrt[3]{x^2-1}+\sqrt[4]{x^3+15}=x^2+2$ I found that $x=1$ is a solution of 
$$\sqrt{x}+\sqrt[3]{x^2-1}+\sqrt[4]{x^3+15}=x^2+2.$$ 
How would one find other real solutions?
 [/INST] To find a polynomial with that root:
You have $A+B+C=x^2+2$.  Consider $(x^2+2)^2=(A+B+C)^2= x+B^2+C^2+2AB+2AC+2BC$.
Take the first 25 powers of $x^2+2$, from $1=(x^2+2)^0$ to $(x^2+2)^{24}$.  They will all be linear combinations of $A,B,C,B^2,C^2,C^3,AB,AB^2,AC,ABC,AB^2C,...$ with coefficients that are polynomials in $x$. But that is a 24-dimensional space, so there is a linear relation between the 25 formulas.  This linear relation is a polynomial in $x$.
</s>