Question

Multiply the polynomials and simplify by combining like terms.
left parenthesis x plus 4 right parenthesis left parenthesis 4 x minus 1 right parenthesis left parenthesis 5 x minus 3 right parenthesis

Answer 1

To multiply the polynomials \((x + 4)(4x - 1)(5x - 3)\), we'll first multiply two of the polynomials and then multiply the result by the third polynomial.

1. **Multiply the first two polynomials**: \((x + 4)(4x - 1)\)

Using the distributive property (FOIL method):

\[
= x \cdot 4x + x \cdot (-1) + 4 \cdot 4x + 4 \cdot (-1)
\]

\[
= 4x^2 - x + 16x - 4
\]

Combine like terms:

\[
= 4x^2 + 15x - 4
\]

2. **Now multiply the result by the third polynomial**: \((4x^2 + 15x - 4)(5x - 3)\)

Again, using the distributive property:

\[
= 4x^2 \cdot 5x + 4x^2 \cdot (-3) + 15x \cdot 5x + 15x \cdot (-3) - 4 \cdot 5x - 4 \cdot (-3)
\]

\[
= 20x^3 - 12x^2 + 75x - 45 - 20x + 12
\]

Combining like terms:

\[
= 20x^3 + (-12x^2) + (75x - 20x) + (-45 + 12)
\]
\[
= 20x^3 - 12x^2 + 55x - 33
\]

Thus, the result of multiplying the given polynomials and simplifying is:

\[
\boxed{20x^3 - 12x^2 + 55x - 33}
\]