Consider the polynomial f(x) = 1 + 2x + 3x² + 4x³. Let s be the sum of all distinct real roots of f(x) and let t = |s|. The area bounded by the curve y = f(x) and the lines x = 0, y = 0 and x = t, lies in the interval (9, 10), (2164, 1116), (0, 2164), (34, 3)?

Let the area be A. We have ∫₀ᵗ f(x) dx ≤ A ≤ ∫₀ᵗ f(x) dx. We have f(x) = 1 + 2x + 3x² + 4x³. The roots of f(x) are difficult to find analytically. Let's analyze the bounds. We know that f(x) is always increasing, so we can estimate the area by considering rectangles.
The integral is ∫₀ᵗ (1 + 2x + 3x² + 4x³) dx = [x + x² + x³ + x⁴]₀ᵗ = t + t² + t³ + t⁴.
We have a cubic equation, so there are at most 3 real roots. Let's try to estimate the sum of roots. Since f(x) is monotonically increasing, it has at most one real root, and it is negative.
Let's approximate the area using the integral:
∫₀ᵗ (1 + 2x + 3x² + 4x³) dx = [x + x² + x³ + x⁴]₀ᵗ = t + t² + t³ + t⁴
Since t is the absolute value of the sum of roots, which is negative, t will be positive.
Let's assume a reasonable value for t, say t = 3. Then the area is approximately 3 + 9 + 27 + 81 = 120. This suggests the interval (2164,1116) might not be correct.
Considering the interval (9,10), the value of t should be around 1. However, this will underestimate the area because f(x) is increasing.
Let's consider numerical approximation. If we approximate the area using rectangles, we can expect it to be roughly between ∫₀ᵗ (1 + 2x + 3x² + 4x³)dx and ∫₀ᵗ (1 + 2x + 3x² + 4x³)dx where the values of the integral represent the lower and upper bounds for the area. The precise calculations would require numerical integration techniques. The given options suggest a numerical approach is necessary. However, without further information or numerical calculations, the precise interval cannot be definitively determined.