Given:
[tex]\[ m(x) = x^2 + 4x \][/tex]

Calculate [tex]\((mn)(x)\)[/tex]:
[tex]\[
\begin{aligned}
(mn)(x) &= (x^2 + 4x)(x) \\
&= x^2 \cdot x + 4x \cdot x \\
&= x^3 + 4x^2
\end{aligned}
\][/tex]

Which is equal to:
A. [tex]\( x^3 + 4x^2 \)[/tex]
B. [tex]\( 5x^2 \)[/tex]
C. [tex]\( 4x^4 \)[/tex]



Answer :

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]