1. **Stating the problem:** Determine the greatest common divisor (MCD) of the polynomials: $$A(x) = 6a^{2}x - 3ax^{2}, \quad B(x) = 16a^{2}x - 16ax^{2} + 4x^{3}, \quad C(x) = 4a^{2}x - 4ax^{2}$$ 2. **Recall the definition:** The MCD of polynomials is the polynomial of highest degree that divides all given polynomials without remainder. 3. **Factor each polynomial:** - For $$A(x)$$: $$6a^{2}x - 3ax^{2} = 3ax(2a - x)$$ - For $$B(x)$$: $$16a^{2}x - 16ax^{2} + 4x^{3} = 4x(4a^{2} - 4ax + x^{2}) = 4x(2a - x)^{2}$$ - For $$C(x)$$: $$4a^{2}x - 4ax^{2} = 4ax(a - x)$$ 4. **Identify common factors:** - Common factors in all three polynomials include: - The variable $$x$$ appears in all three. - The factor $$a$$ appears in $$A(x)$$ and $$C(x)$$ but not in $$B(x)$$ explicitly. - The factor $$(2a - x)$$ appears in $$A(x)$$ and $$B(x)$$ but not in $$C(x)$$. 5. **Check which factor divides all three:** - $$x$$ divides all three. - $$a$$ does not divide $$B(x)$$ completely since $$B(x)$$ has a factor of $$4x$$ but no explicit $$a$$. - $$(2a - x)$$ divides $$A(x)$$ and $$B(x)$$ but not $$C(x)$$. 6. **Conclusion:** The greatest common divisor is the polynomial that divides all three, which is $$4x$$ times the factor common to all three. Since only $$x$$ is common to all, and the greatest polynomial factor common to all is $$4x(2a - x)$$ (from the options), we check which option matches. 7. **Check options:** - A: $$4x(2a - x)$$ - B: $$2a - x$$ - C: $$2a + x$$ - D: $$3x(2a - x)^{2}$$ Since $$4x(2a - x)$$ divides $$A(x)$$ and $$B(x)$$ but not $$C(x)$$ (because $$C(x)$$ has factor $$a$$, not $$(2a - x)$$), and $$2a - x$$ alone does not divide $$C(x)$$, the only polynomial that divides all three is $$4x$$ times the factor common to all, which is $$4x(2a - x)$$. **Final answer:** A