A 2×2 matrix with distinct eigenvalues

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 given by $2x+y=0$, so $v_1=(1,-2)$.

Eigenvector for $\lambda=5$.
$$A-5I = \begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix}.$$
The kernel is given by $-x+y=0$, so $v_2=(1,1)$.

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