To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the two given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] together.
Given:
[tex]\[ f(x) = x + 4 \][/tex]
[tex]\[ g(x) = 3x^2 - 7 \][/tex]
First, we multiply these functions together:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) = (x + 4)(3x^2 - 7) \][/tex]
Now, we need to expand this product step-by-step:
1. Distribute [tex]\( x \)[/tex] across [tex]\( 3x^2 - 7 \)[/tex]:
[tex]\[ x \cdot (3x^2 - 7) = 3x^3 - 7x \][/tex]
2. Distribute [tex]\( 4 \)[/tex] across [tex]\( 3x^2 - 7 \)[/tex]:
[tex]\[ 4 \cdot (3x^2 - 7) = 12x^2 - 28 \][/tex]
3. Combine these results:
[tex]\[ (x + 4)(3x^2 - 7) = 3x^3 - 7x + 12x^2 - 28 \][/tex]
Next, we write it in standard polynomial order, combining like terms where applicable:
[tex]\[ (f \cdot g)(x) = 3x^3 + 12x^2 - 7x - 28 \][/tex]
Thus, the expanded form of [tex]\( (f \cdot g)(x) \)[/tex] is:
[tex]\[ 3x^3 + 12x^2 - 7x - 28 \][/tex]
So, the correct answer is:
[tex]\[ B. \ (f \cdot g)(x) = 3x^3 + 12x^2 - 7x - 28 \][/tex]
