1. **State the problem:** Divide the polynomial $12u^7 + 16u^6 + 4u^5 + 14u^3$ by $2u^4$. 2. **Formula and rules:** When dividing terms with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$ 3. **Divide each term separately:** $$\frac{12u^7}{2u^4} = \frac{12}{2} u^{7-4} = 6u^3$$ $$\frac{16u^6}{2u^4} = \frac{16}{2} u^{6-4} = 8u^2$$ $$\frac{4u^5}{2u^4} = \frac{4}{2} u^{5-4} = 2u$$ $$\frac{14u^3}{2u^4} = \frac{14}{2} u^{3-4} = 7u^{-1} = \frac{7}{u}$$ 4. **Combine all results:** $$6u^3 + 8u^2 + 2u + \frac{7}{u}$$ 5. **Final answer:** $$6u^3 + 8u^2 + 2u + \frac{7}{u}$$