Since the parameter $\sigma$ is positive, the quantity
$$
-2\sigma \left ( rx^2 + y^2 + b(z-r)^2 - br^2 \right )
$$
is negative if
$$
rx^2 + y^2 + b(z-r)^2 > br^{2}.
$$
This inequality defines the exterior of an ellipsoid (call it $E_1$); note that the size of this ellipsoid is fixed by the parameters (that's a hint).
Now the equation
$$
rx^2 + \sigma y^2 + \sigma \left ( z - 2r \right )^2 = C
$$
defines a different ellipsoid, $E_2$, the size of which is determined by your choice of $C$ (another hint).
At this point, remind yourself what it is that you want to show. Typically, the goal is to show that there exists a $C$ such that $E_2$ defines a trapping region for the Lorenz equations, in which case it suffices to show that $E_2$ can be made large enough so that it contains $E_1$. There's really no additional calculation necessary to do this -- you just need to understand what you've done so far.
A related (but different) question is to find an explicit lower bound on $C$ in terms of the parameters. In this case, you can find bounds on each of $x$, $y$, and $z$ separately for points inside of $E_1$. This will then give you a bound on the quantity
$$
rx^2 + \sigma y^2 + \sigma \left ( z - 2r \right )^2
$$
which then defines $C$.