1. **State the problem:** We are given the function $$g(x) = \frac{x^2 - 4x + 8}{3x - 15}$$ and we want to analyze or simplify it. 2. **Identify the formula and rules:** This is a rational function, which is a ratio of two polynomials. Important rules include factoring polynomials to simplify and identifying values that make the denominator zero (which are excluded from the domain). 3. **Factor the denominator:** $$3x - 15 = 3(x - 5)$$ 4. **Check if numerator can be factored:** The quadratic $$x^2 - 4x + 8$$ does not factor nicely over the real numbers because its discriminant is $$\Delta = (-4)^2 - 4 \cdot 1 \cdot 8 = 16 - 32 = -16 < 0$$. 5. **Simplify the function:** Since numerator cannot be factored and denominator factors as above, the function remains: $$g(x) = \frac{x^2 - 4x + 8}{3(x - 5)}$$ 6. **Domain restrictions:** The denominator cannot be zero, so: $$3(x - 5) \neq 0 \implies x \neq 5$$ 7. **Final answer:** $$g(x) = \frac{x^2 - 4x + 8}{3(x - 5)}, \quad x \neq 5$$