Certainly! Let's find [tex]\((c \circ d)(x)\)[/tex], which means we need to evaluate the function [tex]\(c\)[/tex] at [tex]\(d(x)\)[/tex].
Given:
[tex]\[ c(x) = 4x - 2 \][/tex]
and
[tex]\[ d(x) = x^2 + 5x \][/tex]
To find [tex]\((c \circ d)(x)\)[/tex], we need to substitute [tex]\(d(x)\)[/tex] into [tex]\(c(x)\)[/tex]:
[tex]\[ c(d(x)) = c(x^2 + 5x) \][/tex]
Now, substitute [tex]\(x^2 + 5x\)[/tex] in place of [tex]\(x\)[/tex] in the function [tex]\(c(x)\)[/tex]:
[tex]\[ c(x^2 + 5x) = 4(x^2 + 5x) - 2 \][/tex]
Next, distribute the 4 through the polynomial inside the parentheses:
[tex]\[ c(x^2 + 5x) = 4x^2 + 20x - 2 \][/tex]
Therefore, [tex]\((c \circ d)(x)\)[/tex] simplifies to:
[tex]\[ (c \circ d)(x) = 4x^2 + 20x - 2 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{4x^2 + 20x - 2} \][/tex]
