Final answer:
To find the product (f * g)(x) of the functions f(x) = x² + 3x and g(x) = 5x² - 1, you multiply each term of f(x) with each term of g(x) and combine like terms. The final product function is (f * g)(x) = 5x⁴ + 15x³ - x² - 3x.
Step-by-step explanation:
To find the product of the functions f(x) = x² + 3x and g(x) = 5x² - 1, denoted as (f * g)(x), we need to multiply the two functions together. The resulting function (f * g)(x) is calculated by applying the distributive property of multiplication over addition.
(f * g)(x) = f(x) * g(x) = (x² + 3x) * (5x² - 1).
Now expand and simplify:
- x² * 5x² = 5x4
- x² * -1 = -x²
- 3x * 5x² = 15x³
- 3x * -1 = -3x
Combine all the terms:
(f * g)(x) = 5x4 + 15x³ - x² - 3x.
This is the final product of the two functions.