I would first exploit the symmetries of the curve and consider only the first quadrant. Then find the $x$-intercepts to obtain the limits of integration, after which a trigonometric substitution works nicely.
As we can see, there are only even powers of $x$ and $y$, and so we know there is symmetry across both coordinate axes. And so the total area $A$ enclosed will be 4 times the area in the first quadrant. The non-negative $x$-intercepts are found by equating $y$ to zero: