The characteristic equation is \[\det\matrix{-1-\lambda & -4 \\ 2 & -7-\lambda }=0\] We first rewrite the characteristic polynomial of \(A\):
\[ \begin{aligned}
\det(A-\lambda I) = \left\vert \begin{array}{cc} -1-\lambda & -4 \\ 2 & -7-\lambda \end{array} \right\vert &= (-1-\lambda)(-7-\lambda)+4\cdot2 \\
&= (-1-\lambda)(-7-\lambda)+8 \\ &= \lambda^2+8\,\lambda+15
\end{aligned}\] To solve this quadratic equation, we can factor it in the following way: \[ \lambda^2+8\,\lambda+15 = (\lambda+3)(\lambda+5) \] So, the eigenvalues are \(\lambda_1 = -3\) and \(\lambda_2 = -5\).

Let \(\lambda = -3\) be an eigenvalue of the matrix \[A=\matrix{-1 & -4 \\ 2 & -7}\] Then there must be a vector \(\vec{v}\) such that \(A\vec{v}=-3\vec{v}\), this is, \[(A+3 I)\vec{v}=\vec{0}\text.\] In other words, we must find the kernel of the matrix \(A+3 I\). We can do this through row reduction of the matrix \[A+3 I = \matrix{-1 & -4 \\ 2 & -7} - \matrix{-3 & 0 \\0& -3 }=\matrix{ 2 & -4 \\ 2 & -4}\] This can be done as follows:
\[\begin{aligned}
\matrix{2&-4\\2&-4\\}&\sim\matrix{1&-2\\2&-4\\}&{\blue{\begin{array}{c}{{1}\over{2}}R_1\\\phantom{x}\end{array}}}\\\\ &\sim\matrix{1&-2\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2-2R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = -3\) equals \(\left\{ r \cv{-2\\-1} \middle|\;r\in\mathbb R\right\}=\left\langle\cv{-2\\-1}\right\rangle\).

Let \(\lambda = -5\) be an eigenvalue of the matrix \[A=\matrix{-1 & -4 \\ 2 & -7}\] Then there must be a vector \(\vec{v}\) such that \(A\vec{v}=-5\vec{v}\), that is, \[(A+5 I)\vec{v}=\vec{0}\text.\] In other words, we must find the kernel of the matrix \(A+5 I\). We can do this through row reduction of the matrix \[A+5 I = \matrix{-1 & -4 \\ 2 & -7} - \matrix{-5 & 0 \\0& -5 }=\matrix{ 4 & -4 \\ 2 & -2}\] This can be done as follows:
\[\begin{aligned}
\matrix{4&-4\\2&-2\\}&\sim\matrix{1&-1\\2&-2\\}&{\blue{\begin{array}{c}{{1}\over{4}}R_1\\\phantom{x}\end{array}}}\\\\ &\sim\matrix{1&-1\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2-2R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = -5\) 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.