Eigen values are roots of Characteristic equation $|A - \lambda I | = 0.$
For a $3×3$ matrix, characteristic equation will be cubic, so will have $3$ roots. Two roots are given as: $ 2 + i$ and $3$ and We know that complex roots always occur in pairs so, if $2+i$ is a root of characteristic equation, then $2-i$ must be other root.
$\lambda_{1} = 2+i$, $\lambda_{2} = 2-i$ and $\lambda_{3} = 3$
$\color{blue}{\det(A) = \lambda_{1}\lambda_{2}\lambda_{3} = (2+i)*(2-i)*3 = (2^2 - i^2)*3 = 5*3 = 15}$