Irrational Roots

$\text{The constant term of a polynomial is the product of it's roots}\\ \text{So if $\sqrt{2}+\sqrt{5}$ is a root and $c_0$ is rational it must be that$\\ \sqrt{2}-\sqrt{5}$ is also a root}\\ \text{This is where you probably got your idea that $x^2-2x-3$ is part of it$\\$ You made a bit of an error. The actual factor is $x^2-2\sqrt{2}x-3$}\\ \text{Now given that is a factor, and we've got rational coefficients, it must be that $\\x^2 +2\sqrt{2}x-3$ is also a factor}\\ \text{That gets us $\\P(x) = (x^2+2\sqrt{2}x-3)(x^2-2\sqrt{2}x-3) = x^4-14 x^2+9$}\\ P(1)=1-14+9=-4$