Q(x) lies in the third quadrant, increasing on h, while e(x) = Q²(x³+4) is decreasing on (-2, 0).

1. Given the function \( h(x) = f^2(x^3+4) \), we need to explore the monotonicity of this function on the interval \((-2, 0)\).

2. The function \( f(x) \) is increasing in the third quadrant, implying \( f'(x) > 0 \) when \( x \) is in range that places it into that quadrant. However, since specific details about \( f(x) \) are not given, yet it is known to be increasing in a certain quadrant, we consider \( c(x) = x^3 + 4 \).

3. Determine if \( c(x) \) is decreasing over the interval \((-2, 0)\):
- Compute the derivative: \( c'(x) = 3x^2 \).
- At any given \( x \neq 0 \), within the interval \((-2, 0)\), \( 3x^2 > 0 \). This indicates \( c(x) \) is increasing, not decreasing.

4. Since for any composite \( f(g(x)) \) function to be decreasing, derivative of composite function should be less than 0. Thus if \( h(x) \) is assumed to be decreasing, it is feasible only if both derivative conditions clash for \( h'(x) = 2f(x^3 + 4) f'(x) \cdot c'(x) \).

5. Given function \( f \) increases \( \implies \) to make \( h \) definitely decreasing, it should contradict derivative behavior.

6. Conclusion: It suggests evaluation or conditions set might need to verify further parameter specifics of \( f(x) \). However, in principle basis \( f \) expressed differential less likely to turn overall function \( h \) to be negative unless specifics given such allude so.

So the analysis indicates \( x \in (-2, 0) \) where it advises composure for function to situate resolve for warant monotonic check assertions.