1. We are asked to determine the domain of the function $$f(x) = \frac{2x^2 - 6x + 4}{-x^2 - 3x + 10}$$. 2. The domain of a rational function is all real numbers except where the denominator is zero because division by zero is undefined. 3. Set the denominator equal to zero and solve for $x$: $$-x^2 - 3x + 10 = 0$$ 4. Multiply both sides by $-1$ to simplify: $$\cancel{-}x^2 - 3x + 10 = 0 \implies x^2 + 3x - 10 = 0$$ 5. Factor the quadratic: $$x^2 + 3x - 10 = (x + 5)(x - 2) = 0$$ 6. Solve for $x$: $$x + 5 = 0 \implies x = -5$$ $$x - 2 = 0 \implies x = 2$$ 7. These values make the denominator zero, so they must be excluded from the domain. 8. Therefore, the domain of $f$ is all real numbers except $x = -5$ and $x = 2$. 9. In interval notation, the domain is: $$(-\infty, -5) \cup (-5, 2) \cup (2, \infty)$$