Sure, let's work through this problem step-by-step to find [tex]\((m \cdot n)(x)\)[/tex].
We are given two functions:
[tex]\[ m(x) = x^2 + 4x \][/tex]
[tex]\[ n(x) = x \][/tex]
We need to find the product of these two functions, [tex]\((m \cdot n)(x)\)[/tex], which is defined as:
[tex]\[ (m \cdot n)(x) = m(x) \cdot n(x) \][/tex]
To find [tex]\( (m \cdot n)(x) \)[/tex], we multiply [tex]\( m(x) \)[/tex] by [tex]\( n(x) \)[/tex]:
[tex]\[ m(x) \cdot n(x) = (x^2 + 4x) \cdot x \][/tex]
Now let's distribute [tex]\( x \)[/tex] through the terms inside the parentheses:
[tex]\[ (x^2 + 4x) \cdot x = x^2 \cdot x + 4x \cdot x \][/tex]
Utilizing the properties of exponents, we calculate each term:
[tex]\[ x^2 \cdot x = x^{2+1} = x^3 \][/tex]
[tex]\[ 4x \cdot x = 4x^{1+1} = 4x^2 \][/tex]
Now we combine these results:
[tex]\[ x^3 + 4x^2 \][/tex]
Therefore, the product of the functions [tex]\( m(x) \)[/tex] and [tex]\( n(x) \)[/tex] is:
[tex]\[ (m \cdot n)(x) = x^3 + 4x^2 \][/tex]
Thus, the final solution is:
[tex]\[ (m \cdot n)(x) = x^3 + 4x^2 \][/tex]
