Short answer: there is no One True Meaning to the phrase "simplest form"; you'll have to infer from context, or consult your textbook / professor to see if what they want is explicitly stated.
Long answer: it depends on what you're trying to do. For example:
- The first form is clearly simpler if the things you're interested in are things like the roots of the polynomial, or multiplying polynomials.
- The second form is clearly simpler if the things you're interested in are things like the coefficients of your polynomial, or adding polynomials.
Sometimes, completely different things are the simplest form; e.g. sometimes the most useful way to represent a polynomial is by a list of values at various points:
$f(0) = -8$ and $f(1) = -9$ and $f(-1) = -5$
or sometimes you want to write it as a power series about some value other than 0:
$(x+1)^2 - 4(x+1) - 5$
For quadratic polynomials in particular, it is sometimes considered simpler to express them after completing the square
$(x-1)^2 - 9$
or even to write it as a sum or difference of squares:
$(x-1)^2 - 3^2$
Regarding factored form, in some contexts -- e.g. you're mainly doing integer and rational number arithmetic, you would consider
$x^2 - 2$
to already be in factored form. But in other contexts,
$(x - \sqrt{2}) (x + \sqrt{2})$
is the correct factored form.
Hint: recall (ignoring the $x = 2$ case):
$$|x-2| = \begin{cases}
x-2, & x-2 > 0 \\
-(x-2), & x-2 < 0
\end{cases} = \begin{cases}
x-2, & x > 2 \\
-(x-2), & x < 2\text{.}
\end{cases}$$
So $$\dfrac{x^2-x-2}{|x-2|} = \begin{cases}
\dfrac{x^2-x-2}{x-2} = \dfrac{(x-2)(x+1)}{x-2} = x+1, & x > 2 \\
\dfrac{x^2 - x - 2}{-(x-2)} = -(x+1), & x < 2\text{.}
\end{cases}$$
Best Answer
Multiply by $\displaystyle 1 = \frac{2-\sqrt 3}{2-\sqrt 3}$. Use $(a-b)(a+b) = a^2-b^2$ on the denominator. It's called "rationalising the denominator" by multiplying the denominator by the "conjugate surd". You should look up the key phrases in quotes.