First, find the points of intersection by setting the equations equal: \(2x + 1 = -x^2 + 12x - 20\).

Rearrange to get: \(x^2 - 10x + 21 = 0\).

Factorize to find \((x - 3)(x - 7) = 0\), so \(x = 3\) and \(x = 7\).

Calculate the area under the line \(y = 2x + 1\) from \(x = 3\) to \(x = 7\):

Area of trapezium = \(\frac{1}{2} \times 4 \times (7 + 15) = 44\).

Calculate the area under the curve \(y = -x^2 + 12x - 20\) from \(x = 3\) to \(x = 7\):

Integrate to find \(\int_{3}^{7} (-x^2 + 12x - 20) \, dx = \left[ -\frac{x^3}{3} + 6x^2 - 20x \right]_{3}^{7}\).

Evaluate the integral: \(\left[ -\frac{343}{3} + 294 - 140 \right] - \left[ -\frac{27}{3} + 54 - 60 \right] = 54\frac{2}{3}\).

The shaded area is the difference: \(44 - 54\frac{2}{3} = 10\frac{2}{3}\).