3.37: Normal to a curve; AP and GP with shared form

Find the equation of the normal to the curve $y=x^{3}+x^{2}-4x+6$ at the point $(1,4)$.

DO NOT USE A CALCULATOR. Find the exact $x$-coordinates where the normal meets the curve again.

The first three terms of an arithmetic progression are $x$, $5x-4$ and $8x+2$. Find $x$ and the common difference.

A geometric progression has first three terms $y$, $5y-4$ and $8y+2$. Find the two possible values of $y$.

For each $y$, find the corresponding common ratio.