1. **State the problem:** We are given the function $g(x) = (ax^2 + bx + c) \sqrt{-2x + 5}$ and want to understand its domain and behavior. 2. **Formula and rules:** The function involves a square root $\sqrt{-2x + 5}$, which requires the radicand to be non-negative for real values. So, we must have: $$-2x + 5 \geq 0$$ 3. **Solve the inequality for the domain:** $$-2x + 5 \geq 0$$ $$-2x \geq -5$$ Dividing both sides by $\cancel{-2}$ (note the sign change because dividing by a negative): $$x \leq \frac{5}{2}$$ 4. **Domain:** The domain of $g(x)$ is all real $x$ such that: $$x \leq \frac{5}{2}$$ 5. **Behavior:** The polynomial $ax^2 + bx + c$ is defined for all real $x$, so the domain restriction comes solely from the square root. 6. **Summary:** - The function $g(x)$ is defined only for $x \leq \frac{5}{2}$. - For values of $x$ in this domain, $g(x)$ is the product of a quadratic polynomial and the square root of a linear expression. This completes the analysis of the function's domain and form.