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------------------------------------------------ For the functions $ f(x) = 2x^2 - 10x - 10 $ and $ g(x) = 2x^2 - 2x + 7 $, find $ f \cdot g(x) $ and $ f \cdot g(-1) $.

A. $ f \cdot g(x) = 4x^4 - 40x^3 - 60x^2 + 120x + 70 $, $ f \cdot g(-1) = -14 $

B. $ f \cdot g(x) = 4x^4 - 40x^3 + 60x^2 - 120x - 70 $, $ f \cdot g(-1) = 14 $

C. $ f \cdot g(x) = 4x^4 - 40x^3 + 60x^2 - 120x - 70 $, $ f \cdot g(-1) = -14 $

D. $ f \cdot g(x) = 4x^4 - 40x^3 - 60x^2 + 120x + 70 $, $ f \cdot g(-1) = 14 $

The product of functions f(x) = 2x^2 - 10x - 10 and g(x) = 2x^2 - 2x + 7 is f·g(x) = 4x^4 - 40x^3 + 60x^2 - 120x - 70. The value of this function at x=-1 is f·g(-1) = -14.

The correct choice is C.

Explanation:

The question is asking for the product of the two functions f(x) = 2x^2 - 10x - 10 and g(x) = 2x^2 - 2x + 7, which is represented by (f·g(x)), and the value of this resultant function when x=-1, (f·g(-1)).

To find the product of the two functions, multiply them together: (2x^2 - 10x - 10) * (2x^2 - 2x + 7) = 4x^4 - 40x^3 + 60x^2 - 120x - 70. Hence, f·g(x) = 4x^4 - 40x^3 + 60x^2 - 120x - 70.

Then, substitute -1 into the product function to get the resulting value: f·g(-1) = 4*(-1)^4 - 40*(-1)^3 + 60*(-1)^2 - 120*(-1) - 70 = 4 - 40 - 60 + 120 - 70 = -14. Hence, f·g(-1) = -14. Considering both the calculated function and value, the correct answer choice is C: (f·g(x) = 4x^4 - 40x^3 + 60x^2 - 120x - 70), (f·g(-1) = -14)

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