Quartic with a parameter: intercepts, stationary points, tangents and transformations

State, in terms of $a$ where required, the values of $x$ for which $f(x) = 0$.

Find the values of $a$ for which the graph of $y = f(x)$ has

exactly three $x$-intercepts.

exactly four $x$-intercepts.

Let $g$ be the function $g: \R \to \R, g(x) = (x+1)^2(x-2)^2$, which is the function $f$ where $a = 1$.

Find $g'(x)$.

Find the coordinates of the local maximum of $g$.

Find the values of $x$ for which $g'(x) > 0$.

Consider the two tangent lines to the graph of $y = g(x)$ at the points where $x = \frac{-\sqrt{3}+1}{2}$ and $x = \frac{\sqrt{3}+1}{2}$.

Determine the coordinates of the point of intersection of these two tangent lines.