Eigenvalues and eigenvectors of a 2×2 matrix

Characteristic polynomial.
$$\det(A - \lambda I) = \begin{vmatrix} 4-\lambda & 1 \\ 2 & 3-\lambda \end{vmatrix} = (4-\lambda)(3-\lambda) - 2 = \lambda^2 - 7\lambda + 10.$$
The roots are $\lambda_1 = 2$ and $\lambda_2 = 5$.

Eigenvector for $\lambda = 2$.
$$A - 2I = \begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix}.$$
The kernel is spanned by $v_1 = (1,-2)$.

Eigenvector for $\lambda = 5$.
$$A - 5I = \begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix}.$$
The kernel is spanned by $v_2 = (1,1)$.

Conclusion. Eigenvalues $2$ and $5$, with eigenvectors $(1,-2)$ and $(1,1)$ respectively.