1. **State the problem:** Simplify the expression $$\left(\frac{x^3}{y^3}\right) \times \left(\frac{y^8 + x^4}{z^4}\right).$$ 2. **Recall the multiplication rule for fractions:** When multiplying fractions, multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}.$$ 3. **Apply the rule:** Multiply the numerators and denominators: $$\frac{x^3}{y^3} \times \frac{y^8 + x^4}{z^4} = \frac{x^3 \times (y^8 + x^4)}{y^3 \times z^4}.$$ 4. **Distribute the numerator:** $$x^3 \times (y^8 + x^4) = x^3 y^8 + x^3 x^4 = x^3 y^8 + x^{3+4} = x^3 y^8 + x^7.$$ 5. **Write the simplified expression:** $$\frac{x^3 y^8 + x^7}{y^3 z^4}.$$ 6. **Check if further simplification is possible:** We can factor out $x^3$ from the numerator: $$\frac{x^3 (y^8 + x^4)}{y^3 z^4}.$$ This is the simplified form. **Final answer:** $$\boxed{\frac{x^3 (y^8 + x^4)}{y^3 z^4}}.$$