how would you solve:
(t-2)(3t-1)(4t+3)-(2-t)(2-t)(t+5)
To solve the expression (t-2)(3t-1)(4t+3)-(2-t)(2-t)(t+5), we can start by multiplying each pair of binomials within the brackets. Then, we can simplify further by combining like terms and performing any necessary arithmetic operations.
Let's break down the steps:
Step 1: Expand each pair of binomials using the distributive property.
Expanding the first set of binomials:
(t-2)(3t-1)(4t+3) = (t * 3t * 4t) + (t * 3t * 3) + (t * -1 * 4t) + (t * -1 * 3) + (-2 * 3t * 4t) + (-2 * 3t * 3) + (-2 * -1 * 4t) + (-2 * -1 * 3)
Expanding the second set of binomials:
(2-t)(2-t)(t+5) = (2 * 2 * t * (t + 5)) + (2 * 2 * t * -t) + (2 * 2 * t * -5) + (2 * -t * (t + 5)) + (2 * -t * -t) + (2 * -t * -5) + (-t * (t + 5)) + (-t * -t) + (-t * -5)
Step 2: Simplify each expanded expression by combining like terms.
Simplifying the first expanded expression:
(t * 3t * 4t) + (t * 3t * 3) + (t * -1 * 4t) + (t * -1 * 3) + (-2 * 3t * 4t) + (-2 * 3t * 3) + (-2 * -1 * 4t) + (-2 * -1 * 3)
= 12t^3 + 9t^2 - 4t^2 - 3t - 24t^2 - 18t + 8t + 6
= 12t^3 - 19t^2 - 13t + 6
Simplifying the second expanded expression:
(2 * 2 * t * (t + 5)) + (2 * 2 * t * -t) + (2 * 2 * t * -5) + (2 * -t * (t + 5)) + (2 * -t * -t) + (2 * -t * -5) + (-t * (t + 5)) + (-t * -t) + (-t * -5)
= 4t(t + 5) - 4t^2 - 20t - 2t(t + 5) + 2t^2 + 10t + t(t + 5) - t^2 + 5t
= 4t^2 + 20t - 4t^2 - 20t + 2t^2 + 10t + t^2 + 5t^2 + 5t
= 6t^2 + 25t
Step 3: Subtract the simplified second expression from the simplified first expression.
(12t^3 - 19t^2 - 13t + 6) - (6t^2 + 25t)
= 12t^3 - 19t^2 - 13t + 6 - 6t^2 - 25t
= 12t^3 - 19t^2 - 6t^2 - 13t - 25t + 6
= 12t^3 - 25t^2 - 38t + 6
Therefore, the solution to the expression (t-2)(3t-1)(4t+3)-(2-t)(2-t)(t+5) is 12t^3 - 25t^2 - 38t + 6.