Final answer:
The product of the binomials (3x-4)(5-2x)(4x+1) is simplified to -24x^3 + 86x^2 - 57x - 20 through binomial multiplication and combining like terms.
Step-by-step explanation:
To find the product of the binomials (3x-4)(5-2x)(4x+1), we need to perform binomial multiplication step by step. This process involves using the distributive property, commonly known as the FOIL method when dealing with two binomials, and then applying the same strategy to include the third binomial.
First, we multiply the first two binomials:
(3x - 4)(5 - 2x) = 3x(5) + 3x(-2x) - 4(5) - 4(-2x) = 15x - 6x^2 - 20 + 8x
Combine like terms:
15x + 8x - 6x^2 - 20 = -6x^2 + 23x - 20
Next, multiply this result by the third binomial:
(-6x^2 + 23x - 20)(4x + 1) = -6x^2(4x) + -6x^2(1) + 23x(4x) + 23x(1) - 20(4x) - 20(1)
Again, combine like terms:
-24x^3 - 6x^2 + 92x^2 + 23x - 80x - 20 = -24x^3 + 86x^2 - 57x - 20
Thus, the simplified product of the binomials is -24x^3 + 86x^2 - 57x - 20.