Determine whether the system can be stabilized by the control law
Start by computing the closed-loop system. Then compute the characteristic polynomial of the closed-loop system. Finally use the Routh-Hurwitz criterion to see under what conditions of \$G_1\$ and \$G_2\$ the characteristic polynomial will be stable. If no such conditions exist the system cannot be stabilized by the proposed control law.

I did the computations of these using Mathematica and got that the system cannot be stabilized. I will summarize the results, and attach the full computations as a screen shot.

The state matrix of the closed-loop system

$$
\left(
\begin{array}{cccc}
 0 &amp; 0 &amp; 1 &amp; 0 \\
 0 &amp; 0 &amp; 0 &amp; 1 \\
 -G_1 &amp; -G_2-4 &amp; -4 &amp; 0 \\
 4 G_1 &amp; 4 G_2+56 &amp; 16 &amp; 0 \\
\end{array}
\right)
$$ 

The characteristic polynomial
$$
s^4+4 s^3+(G_1-4 G_2 -56) s^2-160 s-40 G_1
$$

The first column of the Routh table
$$
\left(
\begin{array}{c}
 4 \\
 4 G_1-16 G_2-64 \\
 16 \left(160 G_2+640\right) \\
 16 \left(-6400 G_2 G_1-25600 G_1\right) \\
\end{array}
\right)
$$

There exists no conditions under which all of the last three elements are positive.