Let's solve the equation \(\frac{1}{x + 1}\) - \(\frac{1}{x + 3}\) = \(\frac{1}{4}\) step by step:

1. Subtracting the fractions on the left side, we get: \(\frac{(x + 3) - (x + 1)}{(x + 1)(x + 3)}\) = \(\frac{2}{(x + 1)(x + 3)}\) = \(\frac{1}{4}\)

2. Cross-multiplying, we get: \(8 = (x + 1)(x + 3)\)

3. Expanding the right side, we get: \(8 = x^2 + 4x + 3\)

4. Subtracting 8 from both sides, we get: \(0 = x^2 + 4x - 5\)

5. Factoring the right side, we get: 0 = (x + 5)(x - 1)

6. Solving for x, we get: x = -5 or x = 1