The problem defines a binary operation $*$ such that $a * b = a^2 - b^2 + ab$, where $a$ and $b$ are real numbers. We are asked to evaluate $\sqrt{2} * \sqrt{3}$.

1. Problem Description

The problem defines a binary operation $*$ such that $a * b = a^2 - b^2 + ab$, where $a$ and $b$ are real numbers. We are asked to evaluate $\sqrt{2} * \sqrt{3}$.

2. Solution Steps

We are given that $a * b = a^2 - b^2 + ab$. We want to evaluate $\sqrt{2} * \sqrt{3}$.

Let $a = \sqrt{2}$ and $b = \sqrt{3}$. Substituting these values into the formula, we get:

$\sqrt{2} * \sqrt{3} = (\sqrt{2})^2 - (\sqrt{3})^2 + (\sqrt{2})(\sqrt{3})$

$= 2 - 3 + \sqrt{6}$

$= -1 + \sqrt{6}$

3. Final Answer

$\sqrt{2} * \sqrt{3} = -1 + \sqrt{6}$