Maximize square of a quadratic form

Question

Let $A$ be a symmetric $2 \times 2$ real matrix and let $x=(x_1,x_2)^T$ be a vector such that $\|x\| = 1$. We know that the quadratic form $x A x^T$ is maximal if $x$ is parallel to an eigenvector corresponding to the greatest eigenvalue of the matrix $A$. My question is when $(x A x^T)^2$ is maximal?