Final answer:
The factored form of the function f(x) = (x² - 13x + 40)(x² - 2x + 3) is (x - 8)(x - 5)(x² - 2x + 3), after factoring the first quadratic expression into (x - 8)(x - 5) and recognizing that the second expression does not factor over the integers.
Step-by-step explanation:
The factored form of the function f(x) = (x² - 13x + 40)(x² - 2x + 3) can be found by factoring each quadratic expression separately.
First, let's look at the quadratic expression (x² - 13x + 40). To factor this expression, we need to find two numbers that multiply to 40 and add to -13. These numbers are -8 and -5. So we can write (x² - 13x + 40) as (x - 8)(x - 5).
The second quadratic expression (x² - 2x + 3) does not factor neatly over the integers (and is not easily factorable over the real numbers because the discriminant b² - 4ac is negative), so it is already in its simplest form.
Therefore, the factored form of the function is (x - 8)(x - 5)(x² - 2x + 3).