Simplifying rational expressions (ACT). Game on.

Factor: $\dfrac{(x-2)(x-3)}{(x-2)(x+2)}=\dfrac{x-3}{x+2}$ for $x \neq \pm 2\textsf{.}$

Factor: $\dfrac{(x-1)(x+1)}{(x+1)(x+1)}=\dfrac{x-1}{x+1}$ for $x \neq -1\textsf{.}$

Factor: $\dfrac{(x+2)(x+3)}{(x+1)(x+2)}=\dfrac{x+3}{x+1}$ for $x \neq -1$ and $x \neq -2\textsf{.}$

Factor: $\dfrac{(x-4)(x-5)}{(x-3)(x-5)}=\dfrac{x-4}{x-3}$ for $x \neq 3$ and $x \neq 5\textsf{.}$

Factor: $\dfrac{(x-2)(x^{2}+2x+4)}{(x-2)(x+2)}=\dfrac{x^{2}+2x+4}{x+2}$ for $x \neq \pm 2\textsf{.}$

Factor: $\dfrac{(x-1)(x^{2}+x+1)}{(x-1)(x+1)}=\dfrac{x^{2}+x+1}{x+1}$ for $x \neq \pm 1\textsf{.}$

Factor: $\dfrac{2x(x-1)(x+1)}{2x(x-1)}=x+1$ for $x \neq 0$ and $x \neq 1\textsf{.}$

Factor by grouping: $\dfrac{(x^{2}-1)(x+2)}{(x+2)(x-1)}=\dfrac{(x-1)(x+1)(x+2)}{(x+2)(x-1)}=x+1$ for $x \neq -2$ and $x \neq 1\textsf{.}$