A symmetric positive definite matrix inequality

Question

Let $Q$ be a positive definite symmetric matrix. Prove that for any vector $x$, we have $$ \frac{\left(x^{T} x\right)^{2}}{\left(x^{T} Q x\right)\left(x^{T} Q^{-1} x\right)} \geq \frac{4 \lambda_{n} \lambda_{1}}{\left(\lambda_{n}+\lambda_{1}\right)^{2}} $$ where $\lambda_{n}$ and $\lambda_{1}$ are reapectively the largest and smallest eigenvalues of Q.