Final answer:
The student is asked to multiply pairs of functions, f(x) and g(x), then h(x) and k(x), to determine the coefficients of the resulting polynomial. Coefficients are found by performing polynomial multiplication and combining like terms.
Step-by-step explanation:
The student is asking to perform multiplication of two sets of functions and to find the coefficients of the resulting polynomials. In part a, we multiply the functions f(x) = (x^2 + 7x - 12) and g(x) = (x^2 - 9x + 1). In part b, the functions h(x) = (x^2 + 2) and k(x) = (2x^2 - 5x + 7) are multiplied.
For part a, we find the product of f(x) and g(x):
(x^2 + 7x - 12)(x^2 - 9x + 1). This multiplication results in an equation that can be written in the form ax^4 - bx^3 - cx^2 + dx - e, where 'a', 'b', 'c', 'd', and 'e' are the coefficients we need to determine.
For part b, the product of h(x) and k(x) follows a similar process:
(x^2 + 2)(2x^2 - 5x + 7). The resulting polynomial will also be in the form of ax^4 - bx^3 - cx^2 + dx - e.
To find the values of 'a', 'b', 'c', 'd', and 'e', it is necessary to perform polynomial multiplication and combine like terms for each of the function products.