Explanation:

To determine where the function f(x) = x^3 - 3x^2 - 9x + 20 is strictly decreasing, we need to find the intervals where the derivative of the function is negative.

Step 1: Find the derivative of f(x)

To do this, we differentiate the function f(x) with respect to x:

f'(x) = 3x^2 - 6x - 9

Step 2: Solve for f'(x) < />

To find the intervals where f'(x) is negative, we need to solve the inequality f'(x) < />

3x^2 - 6x - 9 < />

Step 3: Solve the quadratic inequality

To solve the quadratic inequality, we can factor it or use the quadratic formula. Factoring the quadratic inequality gives:

(3x + 3)(x - 3) < />

The critical points are x = -1 and x = 3.

Step 4: Create a sign chart

To determine the sign of f'(x) in each interval, we create a sign chart using the critical points and test points.

- We choose a test point for each interval: x = -2, x = 0, and x = 4.

- We plug in the test points into f'(x) to determine the sign of f'(x) in each interval.

Interval 1: x < />
- We plug in x = -2 into f'(x): f'(-2) = 3(-2)^2 - 6(-2) - 9 = 15
- Since f'(-2) > 0, f'(x) is positive in this interval.

Interval 2: -1 < x="" />< />
- We plug in x = 0 into f'(x): f'(0) = 3(0)^2 - 6(0) - 9 = -9
- Since f'(0) < 0,="" f'(x)="" is="" negative="" in="" this="" />

Interval 3: x > 3
- We plug in x = 4 into f'(x): f'(4) = 3(4)^2 - 6(4) - 9 = 15
- Since f'(4) > 0, f'(x) is positive in this interval.

Step 5: Interpret the results

From the sign chart, we can see that f'(x) is negative in the interval -1 < x="" />< 3.="" this="" means="" that="" the="" function="" f(x)="x^3" -="" 3x^2="" -="" 9x="" +="" 20="" is="" strictly="" decreasing="" in="" this="" />

Therefore, the correct answer is option A: (-1, 3).