The characteristic equation is \[\det\matrix{-2-\lambda & -2 \\ 0 & -4-\lambda }=0\] We first rewrite the characteristic polynomial of \(A\):
\[ \begin{aligned}
\det(A-\lambda I) = \left\vert \begin{array}{cc} -2-\lambda & -2 \\ 0 & -4-\lambda \end{array} \right\vert &= (-2-\lambda)(-4-\lambda)+2\cdot0 \\
&= (-2-\lambda)(-4-\lambda)
\end{aligned}\] So, the eigenvalues are \(\lambda_1 = -2\) and \(\lambda_2 = -4\).

Let \(\lambda = -2\) be an eigenvalue of the matrix \[A=\matrix{-2 & -2 \\ 0 & -4}\] Then there must be a vector \(\vec{v}\) such that \(A\vec{v}=-2\vec{v}\), this is, \[(A+2 I)\vec{v}=\vec{0}\text.\] In other words, we must find the kernel of the matrix \(A+2 I\). We can do this through row reduction of the matrix \[A+2 I = \matrix{-2 & -2 \\ 0 & -4} - \matrix{-2 & 0 \\0& -2 }=\matrix{ 0 & -2 \\ 0 & -2}\] This can be done as follows:
\[\begin{aligned}
\matrix{0&-2\\0&-2\\}&\sim\matrix{0&1\\0&-2\\}&{\blue{\begin{array}{c}-{{1}\over{2}}R_1\\\phantom{x}\end{array}}}\\\\ &\sim\matrix{0&1\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2+2R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = -2\) equals \(\left\{ r \cv{1\\0} \middle|\;r\in\mathbb R\right\}=\left\langle\cv{1\\0}\right\rangle\).

Let \(\lambda = -4\) be an eigenvalue of the matrix \[A=\matrix{-2 & -2 \\ 0 & -4}\] Then there must be a vector \(\vec{v}\) such that \(A\vec{v}=-4\vec{v}\), that is, \[(A+4 I)\vec{v}=\vec{0}\text.\] In other words, we must find the kernel of the matrix \(A+4 I\). We can do this through row reduction of the matrix \[A+4 I = \matrix{-2 & -2 \\ 0 & -4} - \matrix{-4 & 0 \\0& -4 }=\matrix{ 2 & -2 \\ 0 & 0}\] This can be done as follows:
\[\begin{aligned}
\matrix{2&-2\\0&0\\}&\sim\matrix{1&-1\\0&0\\}&{\blue{\begin{array}{c}{{1}\over{2}}R_1\\\phantom{x}\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = -4\) equals \(\left\{ r \cv{1\\1} \middle|\;r\in\mathbb R\right\}=\left\langle\cv{1\\1}\right\rangle\).
If possible, we avoided fractions in the solution.