1. **State the problem:** Simplify the expression $\frac{7}{22} \times \frac{96x^3}{8x^2}$.\n\n2. **Write the expression:** $$\frac{7}{22} \times \frac{96x^3}{8x^2}$$\n\n3. **Multiply the numerators and denominators:** $$\frac{7 \times 96x^3}{22 \times 8x^2}$$\n\n4. **Calculate the constants:** $$\frac{672x^3}{176x^2}$$\n\n5. **Simplify the fraction $\frac{672}{176}$ by dividing numerator and denominator by their greatest common divisor (GCD):** The GCD of 672 and 176 is 16. $$\frac{\cancel{16}42}{\cancel{16}11} = \frac{42}{11}$$\n\n6. **Simplify the variables using the rule $\frac{x^a}{x^b} = x^{a-b}$:** $$\frac{x^3}{x^2} = x^{3-2} = x^1 = x$$\n\n7. **Combine the simplified constants and variables:** $$\frac{42}{11} x$$\n\n**Final answer:** $$\frac{42}{11} x$$