Step 1: The total area under the curve is computed as: \[ A = \int_0^{\frac{3}{2}} (1 + 4x - x^2) dx \] Evaluating the integral yields: \[ A = \left[ x + 2x^2 - \frac{x^3}{3} \right]_0^{\frac{3}{2}} = \left[ \frac{3}{2} + 2 \times \left(\frac{3}{2}\right)^2 - \frac{\left(\frac{3}{2}\right)^3}{3} \right] \] The calculated total area is \( \frac{9}{4} \).
Step 2: For the line \(y = mx\) to bisect this area, the area under the line from \(x = 0\) to \(x = \frac{3}{2}\) must equal half of the total area. The area under the line is determined by: \[ A_{{line}} = \int_0^{\frac{3}{2}} mx dx = \frac{3}{2} \times m \times \frac{3}{2} = \frac{9}{4}m \] Setting this equal to half the total area gives: \[ \frac{9}{4}m = \frac{9}{8} \] Solving for \(m\): \[ m = \frac{13}{6} \] Therefore, the value of \(m\) is \( \frac{13}{6} \).