\[ \textbf{Step 1: Define the variables.} \] Let Karan's age be \(K\) and Deepak's age be \(D\). We are given: \[ K \cdot D = 240 \] and \[ 2D = K + 4 \] --- \[ \textbf{Step 2: Express \(K\) in terms of \(D\).} \] \[ K = 2D - 4 \] --- \[ \textbf{Step 3: Substitute into the product equation.} \] \[ (2D - 4) \cdot D = 240 \] \[ 2D^2 - 4D = 240 \] \[ 2D^2 - 4D - 240 = 0 \] \[ D^2 - 2D - 120 = 0 \] --- \[ \textbf{Step 4: Solve the quadratic.} \] \[ D = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-120)}}{2} = \frac{2 \pm \sqrt{484}}{2} = \frac{2 \pm 22}{2} \] \[ D = \frac{24}{2} = 12 \quad \text{or} \quad D = \frac{-20}{2} = -10 \] Since age cannot be negative: \[ D = 12 \] --- \[ \textbf{Step 5: Karan's age.} \] \[ K = 2D - 4 = 2(12) - 4 = 20 \] Check: \[ K \cdot D = 20 \cdot 12 = 240 \quad \checkmark \] --- \[ \textbf{Step 6: Deepak's age two years ago.} \] \[ D - 2 = 12 - 2 = 10 \] --- \[ \boxed{10} \]