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) \( (f \cdot g)(x) = 4x^4 - 20x^3 - 20x^2 + 12x^2 - 60x - 70 \), \( (f \cdot g)(-1) = -110 \)

B) \( (f \cdot g)(x) = 4x^4 - 20x^3 - 20x^2 - 60x - 70 \), \( (f \cdot g)(-1) = -70 \)

C) \( (f \cdot g)(x) = 4x^4 - 12x^3 - 20x^2 + 60x - 70 \), \( (f \cdot g)(-1) = 110 \)

D) \( (f \cdot g)(x) = 4x^4 - 12x^3 - 20x^2 - 60x - 70 \), \( (f \cdot g)(-1) = 70 \)

To find the product of the functions f(x) and g(x), we multiply the two polynomials and simplify the expression. The calculated product function (f⋅g)(x) is 4x⁴ - 24x³ + 14x² + 70x - 70, and the value at x = -1 is (f⋅g)(-1) = -102, which does not match any of the provided options.

Explanation:

To find the product of the functions f(x) and g(x), denoted as (f·g)(x), we need to multiply the functions together. For f(x)=2x²−10x−10 and g(x)=2x²−2x+7, the product is calculated as follows:

(f·g)(x) = (2x²−10x−10)·(2x²−2x+7)

Multiplying the terms, we obtain:

(2x²)(2x²) + (2x²)(−2x) + (2x²)(7) + (−10x)(2x²) + (−10x)(−2x) + (−10x)(7) − 10(2x²) − 10(−2x) − 10(7)

Simplifying the expression:

(f·g)(x) = 4x⁴ - 4x³ + 14x² - 20x³ + 20x² + 70x - 20x² + 20x - 70

We then combine like terms:

(f·g)(x) = 4x⁴ - 24x³ + 14x² + 70x - 70

To find (f·g)(-1), we substitute x with -1:

(f·g)(-1) = 4(-1)⁴ - 24(-1)³ + 14(-1)² + 70(-1) - 70

Simplifying yields:

(f·g)(-1) = 4 - (-24) + 14 - 70 - 70

Calculating the sum:

(f·g)(-1) = 4 + 24 + 14 - 70 - 70 = -102

Therefore, the correct option is not in the provided choices, because the calculation indicates that (f·g)(x) is 4x⁴ - 24x³ + 14x² + 70x - 70 and (f·g)(-1) is -102.