HMMT 2025 A5

Let $\mathcal{S}$ be the set of all nonconstant monic polynomials $P$ with integer coefficients satisfying $P\left(\sqrt{3} + \sqrt{2}\right) = P\left(\sqrt{3} - \sqrt{2}\right)$ . If $Q$ is an element of $\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10) - Q(0)$ .