To solve the problem of finding the composition of the two functions [tex]\( c(x) \)[/tex] and [tex]\( d(x) \)[/tex], let's start by understanding what composition of functions means.
Given:
[tex]\[ c(x) = 4x - 2 \][/tex]
[tex]\[ d(x) = x^2 + 5x \][/tex]
We need to find [tex]\((c \circ d)(x)\)[/tex], which means [tex]\( c(d(x)) \)[/tex].
1. First, substitute [tex]\( d(x) \)[/tex] into [tex]\( c(x) \)[/tex]:
[tex]\[
c(d(x)) = c(x^2 + 5x)
\][/tex]
2. Now, [tex]\( c(x) \)[/tex] states that wherever there is an [tex]\( x \)[/tex] in [tex]\( c(x) \)[/tex], we substitute it with [tex]\( d(x) \)[/tex]:
[tex]\[
c(x^2 + 5x) = 4(x^2 + 5x) - 2
\][/tex]
3. Distribute the [tex]\( 4 \)[/tex] within the parentheses:
[tex]\[
4(x^2 + 5x) = 4x^2 + 20x
\][/tex]
4. Now subtract [tex]\( 2 \)[/tex] from the result:
[tex]\[
4x^2 + 20x - 2
\][/tex]
Thus, the composition function [tex]\((c \circ d)(x) \)[/tex] simplifies to:
[tex]\[
4x^2 + 20x - 2
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[ 4x^2 + 20x - 2 \][/tex]
This corresponds to the fourth choice:
[tex]\[ \boxed{4 x^2 + 20 x - 2} \][/tex]
