1. **State the problem:** Simplify the expression $$\frac{3b^4 c^9 a - 15b^{10} c^2 a + 3b^4 c^5 a^2 + 24b^4 c^3}{3b^4 c^2}$$. 2. **Recall the division rule for exponents:** When dividing terms with the same base, subtract the exponents: $$\frac{x^m}{x^n} = x^{m-n}$$. 3. **Divide each term in the numerator by the denominator separately:** - First term: $$\frac{3b^4 c^9 a}{3b^4 c^2} = \cancel{\frac{3}{3}} \cancel{\frac{b^4}{b^4}} c^{9-2} a = c^7 a$$ - Second term: $$\frac{-15b^{10} c^2 a}{3b^4 c^2} = \frac{-15}{3} b^{10-4} \cancel{\frac{c^2}{c^2}} a = -5 b^6 a$$ - Third term: $$\frac{3b^4 c^5 a^2}{3b^4 c^2} = \cancel{\frac{3}{3}} \cancel{\frac{b^4}{b^4}} c^{5-2} a^2 = c^3 a^2$$ - Fourth term: $$\frac{24b^4 c^3}{3b^4 c^2} = \frac{24}{3} \cancel{\frac{b^4}{b^4}} c^{3-2} = 8 c$$ 4. **Combine all simplified terms:** $$c^7 a - 5 b^6 a + c^3 a^2 + 8 c$$ 5. **Final answer:** $$\boxed{c^7 a - 5 b^6 a + c^3 a^2 + 8 c}$$