√2 + √3 is irrational

Polynomial. If $x=\sqrt2+\sqrt3$, then $x^2=5+2\sqrt6$, so $x^2-5=2\sqrt6$ and $(x^2-5)^2=24$, i.e.
$$x^4-10x^2+1=0.$$

Rational root theorem. Here the polynomial is monic with constant term $1$, so the only rational candidates are $\pm1$. But
$$1-10+1=-8\neq0\quad\text{(for both }x=1\text{ and }x=-1\text{)}.$$
So the polynomial has no rational root. Since $\sqrt2+\sqrt3$ is a root, it is irrational.

Remark. The same method works for many sums of radicals, e.g. $\sqrt2+\sqrt[3]3$.

Lesson. To prove irrationality of an algebraic number, build an integer polynomial it satisfies, then apply the rational root theorem.