1. **State the problem:** Simplify the expression $$(4 - 5x)(1 - 2x)(3x + 2)$$ by multiplying the three binomials. 2. **Use the distributive property (FOIL for binomials) step-by-step:** First, multiply the first two binomials: $$ (4 - 5x)(1 - 2x) = 4 \cdot 1 + 4 \cdot (-2x) - 5x \cdot 1 - 5x \cdot (-2x) $$ 3. **Calculate each term:** $$ = 4 - 8x - 5x + 10x^2 $$ 4. **Combine like terms:** $$ = 4 - 13x + 10x^2 $$ 5. **Now multiply this result by the third binomial $(3x + 2)$:** $$ (4 - 13x + 10x^2)(3x + 2) $$ 6. **Distribute each term:** $$ = 4 \cdot 3x + 4 \cdot 2 - 13x \cdot 3x - 13x \cdot 2 + 10x^2 \cdot 3x + 10x^2 \cdot 2 $$ 7. **Calculate each product:** $$ = 12x + 8 - 39x^2 - 26x + 30x^3 + 20x^2 $$ 8. **Combine like terms:** $$ = 30x^3 + (-39x^2 + 20x^2) + (12x - 26x) + 8 $$ $$ = 30x^3 - 19x^2 - 14x + 8 $$ **Final answer:** $$30x^3 - 19x^2 - 14x + 8$$