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Trying to find the Eigenvalues of this matrix

I have to find the Eigenvalues and Eigenvectors of the following matrix $A$.
$$\begin{pmatrix} 1 & 2 & 1 \\ 1 & 4 & 2 \\ 1 & 2 & 2 \end{pmatrix}$$
I have the following answers and I’m kind of lost on how to prove it.
Eigenvalues:
$ \lambda_1 = 2 – \sqrt{5} $
$ \lambda_2 = 2 + \sqrt{5} $
$ \lambda_3 = -1 $
And I believe the Eigenvectors are $v_1 = (1,2,1), v_2 = (2,4,2), v_3 = (-1,-1,1) $
Is this correct?
Thank you.

A:

We have
$$Av=\lambda v $$
by the properties of linear transformations.
\begin{align}
A\begin{pmatrix}1\\1\\1\end{pmatrix}&=\begin{pmatrix}2-\sqrt5\\1+2\sqrt5\\1\end{pmatrix}\\
&=2\begin{pmatrix}1\\1\\1\end{pmatrix}-\sqrt5\begin{pmatrix}1\\0\\0\end{pmatrix}\\
&=-\sqrt5\begin{pmatrix}1\\0\\0\end{pmatrix}-\sqrt5\begin{pmat
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