1. **State the problem:** Divide the polynomial $4q^3 - 23q^2 - 16q + 67$ by $q - 6$ and express the result including any remainder as a fraction. 2. **Recall the division formula:** For polynomials, dividing $P(q)$ by $D(q)$ gives quotient $Q(q)$ and remainder $R$ such that: $$P(q) = D(q) \times Q(q) + R$$ where $\deg(R) < \deg(D)$. 3. **Set up synthetic division:** Since divisor is $q - 6$, use $6$ for synthetic division. 4. **Coefficients of dividend:** $4, -23, -16, 67$ 5. **Perform synthetic division:** - Bring down 4. - Multiply 4 by 6 = 24; add to -23 = 1. - Multiply 1 by 6 = 6; add to -16 = -10. - Multiply -10 by 6 = -60; add to 67 = 7. 6. **Quotient and remainder:** Quotient coefficients are $4, 1, -10$ corresponding to $4q^2 + q - 10$, remainder is 7. 7. **Express final answer:** $$\frac{4q^3 - 23q^2 - 16q + 67}{q - 6} = 4q^2 + q - 10 + \frac{7}{q - 6}$$