Find a possible function $h\left(x\right)$ and a number $c$ such that the following limit represents the slope of the curve $y=h\left(x\right)$ at $(c,h(c\left)\right)$. Then, evaluate the limit:

$\underset{x\to -1}{\lim }\frac{4x^2-2x-6}{x+1}$

Determine the derivative of the function $g\left(t\right)=5t^2+2t$ using limits.

The following formulas for $f_{-}^{\prime}\left(a\right)$ and $f_{+}^{\prime}\left(a\right)$ represent the left- and right-sided derivatives of a function at a point $a$, respectively:

$f_{-}^{\prime}\left(a\right)={\displaystyle\lim_{h\to0^{-}}{\frac{f(a+h)-f(a)}{h}}}$, $f_{+}^{\prime}\left(a\right)={\displaystyle\lim_{h\to0^{+}}{\frac{f(a+h)-f(a)}{h}}}$

Consider $f\left(x\right)=\{\begin{array}{c}6-x^2\text{ if }x\le 2\\ 3x-4\text{ if }x>2\end{array}$. Find $f_{-}^{\prime}\left(a\right)$ and $f_{+}^{\prime}\left(a\right)$ at $a=2$.