For the given functions f(x) and g(x), find (f ∙ g)(x) or $ \left(\frac{f}{g}\right)(x) $ as indicated.
-$$f ( x ) = 5 x ^ { 2 } + 4 x + 2 , g ( x ) = x ^ { 2 } + 4 x + 4$$
Find $( f \cdot g ) ( x )$.
A) $5 x ^ { 4 } + 24 x ^ { 3 } + 38 x ^ { 2 } + 24 x + 8$
B) $5 x ^ { 4 } + 20 x ^ { 3 } + 36 x ^ { 2 } + 24 x + 8$
C) $5 x ^ { 4 } + 20 x ^ { 3 } + 38 x ^ { 2 } + 24 x + 8$
D) $5 x ^ { 4 } + 24 x ^ { 3 } + 36 x ^ { 2 } + 24 x + 8$

Question

Quizplus · Accepted Answer

To find $(f \cdot g)(x)$, multiply $f(x) = 5x^2 + 4x + 2$ by $g(x) = x^2 + 4x + 4$. This results in $5x^4 + 20x^3 + 16x^3 + 8x^2 + 16x^2 + 24x + 8$, which simplifies to $5x^4 + 36x^3 + 24x^2 + 24x + 8$. However, the correct simplification should be $5x^4 + 24x^3 + 38x^2 + 24x + 8$, indicating a mistake in my initial explanation. The correct process involves multiplying each term in $f(x)$ by each term in $g(x)$ and then combining like terms.

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