Step-by-step Solution:

Here is the MathJax code for the given solution: \[ \text{Given: Second-degree equation, } 2x^2 + 7xy + 3y^2 + 8x + 14y + \lambda = 0 \] \[ \text{Compare with the general second-degree equation } ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 \] \[ \text{So, } a = 2, \quad b = 3, \quad h = \frac{7}{2}, \quad g = 4, \quad f = 7, \quad c = \lambda \] \[ \text{Given that the equation represents a pair of straight lines:} \] \[ \Delta = abc + 2fgh - a f^2 - b g^2 - ch^2 = 0 \] \[ \Rightarrow 2 \times 3 \times \lambda + 2 \times 7 \times 4 \times \frac{7}{2} - 2 \times (7)^2 - 3 \times (4)^2 - \lambda \times \left(\frac{7}{2}\right)^2 = 0 \] \[ \Rightarrow 6\lambda + 196 - 98 - 48 - \frac{49\lambda}{4} = 0 \] \[ \Rightarrow 50 - \frac{25\lambda}{4} = 0 \] \[ \Rightarrow 200 - 25\lambda = 0 \] \[ \therefore \lambda = 8 \]