1. **State the problem:** Simplify the expression $$\frac{3x^3 - 50x + 8}{3x^2 + 12x + 2}$$. 2. **Factor the denominator:** The denominator is a quadratic expression: $$3x^2 + 12x + 2$$ We look for factors or use the quadratic formula: $$x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{-12 \pm \sqrt{144 - 24}}{6} = \frac{-12 \pm \sqrt{120}}{6} = \frac{-12 \pm 2\sqrt{30}}{6} = -2 \pm \frac{\sqrt{30}}{3}$$ Since it does not factor nicely, we keep it as is. 3. **Try to factor the numerator:** $$3x^3 - 50x + 8$$ We attempt to factor by grouping or find rational roots using the Rational Root Theorem. Possible roots are factors of 8 over factors of 3: $$\pm1, \pm2, \pm4, \pm8, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{4}{3}, \pm\frac{8}{3}$$. 4. **Test possible roots:** Test $x=2$: $$3(2)^3 - 50(2) + 8 = 3(8) - 100 + 8 = 24 - 100 + 8 = -68 \neq 0$$ Test $x=1$: $$3(1)^3 - 50(1) + 8 = 3 - 50 + 8 = -39 \neq 0$$ Test $x=4$: $$3(4)^3 - 50(4) + 8 = 3(64) - 200 + 8 = 192 - 200 + 8 = 0$$ So, $x=4$ is a root. 5. **Divide numerator by $(x-4)$:** Using synthetic division: Coefficients: 3 (for $x^3$), 0 (for $x^2$), -50 (for $x$), 8 (constant) Bring down 3. Multiply 3 by 4 = 12, add to 0 = 12. Multiply 12 by 4 = 48, add to -50 = -2. Multiply -2 by 4 = -8, add to 8 = 0. So quotient is: $$3x^2 + 12x - 2$$ 6. **Rewrite numerator:** $$3x^3 - 50x + 8 = (x - 4)(3x^2 + 12x - 2)$$ 7. **Simplify the expression:** $$\frac{(x - 4)(3x^2 + 12x - 2)}{3x^2 + 12x + 2}$$ 8. **Check if numerator and denominator share factors:** Denominator is $$3x^2 + 12x + 2$$ Numerator factor is $$3x^2 + 12x - 2$$ They differ only in the constant term, so no common factors. 9. **Final simplified form:** $$\frac{(x - 4)(3x^2 + 12x - 2)}{3x^2 + 12x + 2}$$ This is the simplest factorization form. **Answer:** $$\boxed{\frac{(x - 4)(3x^2 + 12x - 2)}{3x^2 + 12x + 2}}$$