Answer:
(fg)(x) = x^4 - x^3 + 15x^2 - 6x + 54
Explanation:
We want to multiply and simplify as much as possible:
f(x) * g(x)
(x^2 + 6)(x^2 - x + 9)
(x^2 * x^2) + (x^2 * - x) + (x^2 * 9) + (6 * x^2) + (6 * - x) + (6 * 9)
Note that when you're multiplying exponents, we add them:
x^4 + - x^3 + 9x^2 + 6x^2 - 6x + 54
Now we add 9x^2 and 6x^2 as they are like terms:
x^4 - x^3 + 15x^2 - 6x + 54
Thus, (fg)(x) simplified is x^4 - x^3 + 15x^2 - 6x + 54.
Optional: Check the validity of the answer:
We can check that our answer is correct by plugging in a number for x in both the unsimplified and simplified expression and seeing if we get the same answer. Let's try 5:
Plugging in 5 for x in (x^2 + 6)(x^2 - x + 9):
(5^2 + 6)(5^2 - 5 + 9)
(25 + 6)(25 - 5 + 9)
(31)(20 + 9)
(31)(29)
899
Plugging in 5 for x in x^4 - x^3 + 15x^2 - 6x + 54:
5^4 - (5)^3 + 15(5)^2 - 6(5) + 54
625 - 125 + 15(25) - 30 + 54
625 - 125 + 375 - 30 + 54
500 + 375 - 30 + 54
875 - 30 + 54
845 + 54
899
Thus, our answer is correct.