To find the expression equivalent to [tex]\((m n)(x)\)[/tex] where [tex]\(m(x) = x^2 + 3\)[/tex] and [tex]\(n(x) = 5x + 9\)[/tex], you need to understand that [tex]\((m n)(x)\)[/tex] represents the product of the two functions [tex]\(m(x)\)[/tex] and [tex]\(n(x)\)[/tex]. That means:
[tex]\[
(m n)(x) = m(x) \cdot n(x)
\][/tex]
Substitute the expressions for [tex]\(m(x)\)[/tex] and [tex]\(n(x)\)[/tex]:
[tex]\[
m(x) = x^2 + 3
\][/tex]
[tex]\[
n(x) = 5x + 9
\][/tex]
Now, we want to multiply these two expressions:
[tex]\[
(x^2 + 3)(5x + 9)
\][/tex]
To multiply these expressions, we'll use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(x^2 + 3)(5x + 9) = x^2 \cdot 5x + x^2 \cdot 9 + 3 \cdot 5x + 3 \cdot 9
\][/tex]
Let's calculate each term step by step:
[tex]\[
x^2 \cdot 5x = 5x^3
\][/tex]
[tex]\[
x^2 \cdot 9 = 9x^2
\][/tex]
[tex]\[
3 \cdot 5x = 15x
\][/tex]
[tex]\[
3 \cdot 9 = 27
\][/tex]
Now, combine all the terms:
[tex]\[
(m n)(x) = 5x^3 + 9x^2 + 15x + 27
\][/tex]
So, the expression equivalent to [tex]\((m n)(x)\)[/tex] is:
[tex]\[
5x^3 + 9x^2 + 15x + 27
\][/tex]
Thus, the correct answer is:
[tex]\[
5x^3 + 9x^2 + 15x + 27
\][/tex]
