Expand Simplify C4061C

1. **State the problem:** Expand and simplify the expression $$(t - 2)(t + 5)(t - 4)$$. 2. **Use the distributive property (FOIL) step-by-step:** First, multiply the first two binomials: $$ (t - 2)(t + 5) = t \cdot t + t \cdot 5 - 2 \cdot t - 2 \cdot 5 = t^2 + 5t - 2t - 10 = t^2 + 3t - 10 $$ 3. **Now multiply the result by the third binomial:** $$ (t^2 + 3t - 10)(t - 4) $$ 4. **Distribute each term:** $$ t^2 \cdot t = t^3 $$ $$ t^2 \cdot (-4) = -4t^2 $$ $$ 3t \cdot t = 3t^2 $$ $$ 3t \cdot (-4) = -12t $$ $$ -10 \cdot t = -10t $$ $$ -10 \cdot (-4) = 40 $$ 5. **Combine all terms:** $$ t^3 - 4t^2 + 3t^2 - 12t - 10t + 40 $$ 6. **Simplify like terms:** $$ t^3 + (-4t^2 + 3t^2) + (-12t - 10t) + 40 = t^3 - t^2 - 22t + 40 $$ **Final answer:** $$ t^3 - t^2 - 22t + 40 $$