To solve the given problem step-by-step, start by defining the functions involved:
1. Let [tex]\( m(x) = x^2 + 4x \)[/tex].
We need to find the expression for [tex]\((m \cdot n)(x)\)[/tex], where [tex]\( n(x) = x \)[/tex]. In other words, we are looking to multiply the function [tex]\( m(x) \)[/tex] by [tex]\( n(x) \)[/tex].
2. Calculate [tex]\((m \cdot n)(x)\)[/tex]:
[tex]\[
(m \cdot n)(x) = m(x) \cdot n(x)
\][/tex]
Given [tex]\( n(x) = x \)[/tex], substitute this into the multiplication:
[tex]\[
(m \cdot n)(x) = m(x) \cdot x
\][/tex]
3. Now substitute [tex]\( m(x) \)[/tex] as [tex]\( x^2 + 4x \)[/tex]:
[tex]\[
(m \cdot n)(x) = (x^2 + 4x) \cdot x
\][/tex]
4. Distribute [tex]\( x \)[/tex] across the terms inside the parentheses:
[tex]\[
(x^2 + 4x) \cdot x = x^2 \cdot x + 4x \cdot x
\][/tex]
5. Simplify each term by performing the multiplications:
[tex]\[
x^2 \cdot x = x^3
\][/tex]
[tex]\[
4x \cdot x = 4x^2
\][/tex]
6. Combine the simplified terms:
[tex]\[
x^3 + 4x^2
\][/tex]
So, [tex]\((m \cdot n)(x) = x^3 + 4x^2\)[/tex].
Hence, the correct expression is:
[tex]\[
x^3 + 4x^2
\][/tex]
The given question's options were:
- [tex]\( x^3 + 4x^2 \)[/tex]
- [tex]\( 5x^2 \)[/tex]
- [tex]\( 4x^4 \)[/tex]
From the calculations, we can conclude that the correct answer is:
[tex]\[
x^3 + 4x^2
\][/tex]
