1. **State the problem:** Find the inverse Laplace transform of the function $$F(s) = \frac{2s - 3}{s^2 - 4}$$. 2. **Factor the denominator:** Note that $$s^2 - 4 = (s - 2)(s + 2)$$. 3. **Partial fraction decomposition:** Express $$F(s)$$ as $$\frac{2s - 3}{(s - 2)(s + 2)} = \frac{A}{s - 2} + \frac{B}{s + 2}$$ Multiply both sides by $$ (s - 2)(s + 2) $$: $$2s - 3 = A(s + 2) + B(s - 2)$$ 4. **Find coefficients A and B:** Set $$s = 2$$: $$2(2) - 3 = A(2 + 2) + B(0) \Rightarrow 4 - 3 = 4A \Rightarrow 1 = 4A \Rightarrow A = \frac{1}{4}$$ Set $$s = -2$$: $$2(-2) - 3 = A(0) + B(-2 - 2) \Rightarrow -4 - 3 = -4B \Rightarrow -7 = -4B \Rightarrow B = \frac{7}{4}$$ 5. **Rewrite the function:** $$F(s) = \frac{1/4}{s - 2} + \frac{7/4}{s + 2}$$ 6. **Recall inverse Laplace transforms:** $$\mathcal{L}^{-1}\left\{ \frac{1}{s - a} \right\} = e^{at}$$ 7. **Apply inverse Laplace transform:** $$f(t) = \frac{1}{4}e^{2t} + \frac{7}{4}e^{-2t}$$ **Final answer:** $$\boxed{f(t) = \frac{1}{4}e^{2t} + \frac{7}{4}e^{-2t}}$$