Double Simplicity

Algebra Level 5

Let $x,y$ be real numbers such that $x^2 + xy + 2y^2 = 8$ .

The greatest value that $x + y$ can attain is a real number equal to $\dfrac{a\sqrt{b}}{b}$ , where $a,b$ are positive integers such that $b$ does not divide $a$ , and $b$ is square- free.

Evaluate $a+b$ .