Asymptotical stability of the origin of a non linear system

Question

Let the system $\begin{cases} \dot x_1=-x_1-x_1^3+x_2\\ \dot x_2=-x_1-x_2 \end{cases}$
The origin is in my opinion asymptotically stable since:
let $V(x)=\frac{1}{2}(x_1^2+x_2^2)$, that is positive definite and radially unbounded, then $\dot V(x)=-x_1^2-x_2^2-x_2^4=-x^Tx-x_1^4$, that is negative definite, it is right?
In particular my doubt: it is true to think that $-x^Tx-x_1^4$ gives me a negative definite $\dot V$?
Moreover if I consider that $-x^Tx-x_1^4\leq-x^Tx$, so can I conclude that in addition $x=0$ is exponentially stable?