linear algebra - Characteristic polynomial proof

The trace of a matrix is the sum of the entries on its main diagonal. Prove that if $A$ is a $2 \times 2$ matrix, then the characteristic polynomial of $A$ is $x^2 − {c_1}x + c_2$ where $c_1$ is the trace of $A$ and $c_2$ is the determinant of $A$.

Can anyone explain this to me? So far, I only know that $C_a (x) = \operatorname{det}(A-xI)$, that the product of eigenvalues (counting multiplicity) is the $\operatorname{det}{A}$, and the sum of eigenvalues (counting multiplicity) equals the trace of $A$. I am just lost as to how to apply these.