3.1: Cubic factor, curve intersections and shaded area

Find $p(x)$ in the form $(2x - 1)q(x)$, where $q(x)$ is a quadratic factor.

The diagram shows the graph of $y = \dfrac{1}{x}$ for $x > 0$, and the graph of $y = -2x^{2} + 3x + 1$. The curves intersect at the points $A$ and $B$.

Using your answer to part (a), find the exact $x$-coordinate of $A$ and of $B$.

Find the exact area of the shaded region.