Proof that the characteristic polynomial of a matrix can be expressed as a sum of the matrix's determinant and scaled powers of its eigen values

Question

In the book, Introduction to Computational Linear Algebra, the authors state the following:

For any matrix $A \in \mathbb{R}^n$, its characteristic polynomial, $p_\text{A}(\lambda) = \text{det}(A - \lambda I)$ can be expressed as

$p_\lambda(A) = |A| + \sum_{i=1}^n c_i\lambda^i$, where $c_i \in \mathbb{R}$ and $i = 1, 2, \dots , n$.

I don't understand how the above equation was derived. Can someone please help me with the proof?