To simplify the given expression \( Q = \left(\frac{4x}{x+2} - \frac{8x^2}{x^2-4}\right) \div \left(3 + \frac{x+2}{x-2}\right) \), let's break it down step by step.

First, simplify the individual components of the expression.

### Step 1: Simplify the first part of the numerator
\[ \frac{4x}{x+2} - \frac{8x^2}{x^2-4} \]

Notice that \( x^2 - 4 \) can be factored as \( (x+2)(x-2) \):
\[ \frac{4x}{x+2} - \frac{8x^2}{(x+2)(x-2)} \]

To combine these fractions, find a common denominator, which is \( (x+2)(x-2) \):
\[ \frac{4x(x-2)}{(x+2)(x-2)} - \frac{8x^2}{(x+2)(x-2)} \]

Combine the fractions:
\[ \frac{4x(x-2) - 8x^2}{(x+2)(x-2)} \]

Simplify the numerator:
\[ 4x(x-2) - 8x^2 = 4x^2 - 8x - 8x^2 = -4x^2 - 8x \]

So, the simplified form of the first part is:
\[ \frac{-4x^2 - 8x}{(x+2)(x-2)} = \frac{-4x(x + 2)}{(x+2)(x-2)} \]

Cancel out the common factor \( x+2 \):
\[ \frac{-4x}{x-2} \]

### Step 2: Simplify the denominator
\[ 3 + \frac{x+2}{x-2} \]

Combine into a single fraction:
\[ 3 + \frac{x+2}{x-2} = \frac{3(x-2) + (x+2)}{x-2} = \frac{3x - 6 + x + 2}{x-2} = \frac{4x - 4}{x-2} \]

Factor out the common factor in the numerator:
\[ \frac{4(x-1)}{x-2} \]

### Step 3: Combine the simplified parts
Now we have:
\[ Q = \frac{\frac{-4x}{x-2}}{\frac{4(x-1)}{x-2}} \]

This simplifies to:
\[ Q = \frac{-4x}{x-2} \times \frac{x-2}{4(x-1)} \]

Cancel out the common factor \( x-2 \):
\[ Q = \frac{-4x}{4(x-1)} = \frac{-x}{x-1} \]

So, the simplified form of the expression \( Q \) is:
\[ Q = \frac{-x}{x-1} \]