Math Snap

STEP 1

Assumptions1. We have two functions, $c(x)=4x-$ and $d(x)=x^{}+5x$.
. We need to find the product of these two functions, denoted as $(c \cdot d)(x)$.

STEP 2

The product of two functions $c(x)$ and $d(x)$ is defined as $(c \cdot d)(x) = c(x) \cdot d(x)$.

STEP 3

Substitute the given functions into the product.
$$(c \cdot d)(x) = (x-2) \cdot (x^{2}+5x)$$

STEP 4

To multiply these two expressions, we use the distributive property of multiplication over addition, which states that $a \cdot (b + c) = a \cdot b + a \cdot c$.
(c \cdot d)(x) =4x \cdot x^{2} +4x \cdotx -2 \cdot x^{2} -2 \cdotx

STEP 5

implify the above expression.
$$(c \cdot d)(x) =4x^{3} +20x^{2} -2x^{2} -10x$$

STEP 6

Combine like terms.
$$(c \cdot d)(x) =4x^{3} + (20x^{2} -2x^{2}) -10x$$

Calculate the final expression.
$$(c \cdot d)(x) =4x^{3} +18x^{2} -10x$$So, $(c \cdot d)(x) =4x^{3} +18x^{2} -10x$.