1. The given function is $$G(x) = (7x + 3)^4 (5x - 2)(3x + 2)^4 (5x + 3)^4 (5x - 1).$$ 2. We are told the multiplicity of the zero $$-\frac{3}{5}$$ of $$G$$ is $$\frac{3}{7} m$$, and we need to find $$m$$. 3. First, find which factor corresponds to the zero $$-\frac{3}{5}$$. 4. Set each factor to zero to find zeros: - For $$7x + 3 = 0$$, $$x = -\frac{3}{7}$$ - For $$5x - 2 = 0$$, $$x = \frac{2}{5}$$ - For $$3x + 2 = 0$$, $$x = -\frac{2}{3}$$ - For $$5x + 3 = 0$$, $$x = -\frac{3}{5}$$ - For $$5x - 1 = 0$$, $$x = \frac{1}{5}$$ 5. The zero $$-\frac{3}{5}$$ corresponds to the factor $$(5x + 3)^4$$. 6. Its multiplicity is the exponent 4. 7. Since multiplicity is given by $$\frac{3}{7} m$$, set equal to 4: $$\frac{3}{7} m = 4$$ 8. Solve for $$m$$: $$m = 4 \times \frac{7}{3} = \frac{28}{3}$$ Final answer: $$m = \frac{28}{3}$$