Short Solution:

Let the two numbers be \( a \) and \( b \).

Given: Harmonic mean is 4: $$\frac{2ab}{a + b} = 4 \quad \text{(1)}$$ Arithmetic mean \( A = \frac{a + b}{2} \),
Geometric mean \( G = \sqrt{ab} \)

Given: $$2A + G^2 = 27$$ $$2 \cdot \frac{a + b}{2} + ab = 27 \Rightarrow a + b + ab = 27 \quad \text{(2)}$$ From (1): Multiply both sides by \( a + b \): $$2ab = 4(a + b) \Rightarrow ab = 2(a + b) \quad \text{(3)}$$ Substitute (3) into (2): $$a + b + 2(a + b) = 27 \Rightarrow 3(a + b) = 27 \Rightarrow a + b = 9$$ Then from (3): $$ab = 2 \cdot 9 = 18$$ Now solve: $$x^2 - (a + b)x + ab = 0 \Rightarrow x^2 - 9x + 18 = 0$$ $$\Rightarrow x = 3, 6$$
Final Answer:
$$\boxed{3 \text{ and } 6}$$