The tangent to the curve

at the point $P$ intersects the curve again at the point $Q.$

1. Given that $P$ has $x$-coordinate $-1,$ show that $Q$ has $x$-coordinate $1$ and find the $y$ coordinate of $Q.$

2. Prove that the tangent does not intersect the curve for a third time.

Find the equation of the tangent at the point $P.$

You will need to expand the cubic in order to differentiate it.

In the end, you should find yourself solving the equation

It is possible to guess the solutions to this equation.

For part (b), you know that $(x-1)$ and $(x+1)$ are both roots of

Use long division to factorise this cubic and hence prove that the only two solutions represent the $x$-coordinates of $P$ and $Q.$