1. **State the problem:** We are given the equation $$\frac{-2x - 12}{x (x - 2) (x + 2)} = \frac{A}{x} + \frac{-2}{x - 2} + \frac{-1}{x + 2}$$ and need to find the value of $A$. 2. **Use the formula for partial fraction decomposition:** The expression on the left is decomposed into partial fractions on the right. Each term corresponds to one factor in the denominator. 3. **Multiply both sides by the common denominator** $x (x - 2) (x + 2)$ to clear fractions: $$-2x - 12 = A (x - 2)(x + 2) + (-2) x (x + 2) + (-1) x (x - 2)$$ 4. **Expand each term:** - $A (x - 2)(x + 2) = A (x^2 - 4)$ - $-2 x (x + 2) = -2 (x^2 + 2x) = -2x^2 - 4x$ - $-1 x (x - 2) = -1 (x^2 - 2x) = -x^2 + 2x$ 5. **Combine all terms on the right:** $$-2x - 12 = A x^2 - 4A - 2x^2 - 4x - x^2 + 2x$$ 6. **Group like terms:** $$-2x - 12 = (A x^2 - 2x^2 - x^2) + (-4x + 2x) - 4A$$ $$-2x - 12 = (A - 3) x^2 - 2x - 4A$$ 7. **Match coefficients of powers of $x$ on both sides:** - Coefficient of $x^2$: Left side has 0, so $A - 3 = 0$ which gives $A = 3$ - Coefficient of $x$: Left side is $-2$, right side is $-2$, consistent. - Constant term: Left side is $-12$, right side is $-4A = -4(3) = -12$, consistent. **Final answer:** $$\boxed{3}$$