1. **State the problem:** Factor the polynomial $4x^3 + 6x^2 - 18x$ completely. 2. **Identify the greatest common factor (GCF):** Look at the coefficients 4, 6, and -18. The GCF of 4, 6, and 18 is 2. Also, each term contains at least one $x$, so the GCF includes $x$. 3. **Factor out the GCF:** $$4x^3 + 6x^2 - 18x = 2x(2x^2 + 3x - 9)$$ 4. **Factor the quadratic inside the parentheses:** We need to factor $2x^2 + 3x - 9$. 5. **Use the AC method:** Multiply $a=2$ and $c=-9$ to get $-18$. Find two numbers that multiply to $-18$ and add to $3$: these are $6$ and $-3$. 6. **Rewrite the middle term:** $$2x^2 + 6x - 3x - 9$$ 7. **Group terms:** $$(2x^2 + 6x) + (-3x - 9)$$ 8. **Factor each group:** $$2x(x + 3) - 3(x + 3)$$ 9. **Factor out the common binomial:** $$(2x - 3)(x + 3)$$ 10. **Write the complete factorization:** $$4x^3 + 6x^2 - 18x = 2x(2x - 3)(x + 3)$$ **Final answer:** $2x(2x - 3)(x + 3)$