Find the product of the following polynomials:
-4x(21x^2-3x+2)
-13x(7x^3-11x+5)
(3b+2)(4b-5)
(4x-9)^2

A)  -84x^3 - 8x

B)  -91x^4 + 143x^2 - 65x

C)  12b^2 - 7b - 10.

D)  16x^2 - 72x + 81

Step-by-step explanation:

A) -4x(21x^2-3x+2)  

B) -13x(7x^3-11x+5)

C) (3b+2)(4b-5)

D) (4x-9)^2

In A) -4x(21x^2-3x+2)  we are multiplying the binomial (21x^2-3x+2)  by the monomial -4x; there are two multiplications involved:  

-4x(21x^2) = -84x^3

and

-4x(-3x+2) = +12x^2 - 8x.

Hence A) -4x(21x^2-3x+2)   = -84x^3 - 8x

B)  The work done to find the product in B) is similar:  Multiply each term in 7x^3-11x+5 by -13x:

The end result is -91x^4 + 143x^2 - 65x

C) Here we are multiplying together two binomials; we use the FOIL method:  Multiply together the First terms, then the Outer terms, then the Inner terms, and finally the Last terms.  This results in:

(3b+2)(4b-5)  = 12b^2 -15b + 8b -10, or, after simplification, 12b^2 - 7b - 10.

In D) we are squaring a binomial.  The formula for this is:

(a - b)^2 = a^2 - 2ab + b^2.  Here,

(4x - 9)^2 = 16x^2 - 2(36x) + 81, or 16x^2 - 72x + 81