3.16: Shaded cubic region; AP 12th term; GP equation

The curve $y=x^{3}+2x^{2}-2x$ has point $A$ with $x$-coordinate $1.5$. The tangent at $A$ meets the $x$-axis at $B$; the curve meets the $x$-axis at $C$. Find the area of the shaded region.

For an AP, $S_{20}=1100$ and $S_{70}=14350$. Find the $12$th term.

A GP has first three terms $x+6$, $9-x$, $\dfrac{1}{x+1}$. Show that $x$ satisfies $x^{2}-43x+156=0$, and hence show a sum to infinity exists for each possible $x$.