Factor the numerator by taking out the common factor of -3.

We need two numbers that multiply to 18 and add to -9. These numbers are -3 and -6.

$$a^2 - 9a + 18 = (a - 3)(a - 6)$$

$$\frac{-3(a - 6)}{(a - 3)(a - 6)} = \frac{-3}{a - 3}$$

The simplified expression is $\boxed{\frac{-3}{a - 3}}$.

$4x^2 - 1$ is a difference of squares, $(2x)^2 - 1^2$.

$$4x^2 - 1 = (2x - 1)(2x + 1)$$

$$\frac{(2x - 1)(2x + 1)}{4x(x - 2)}$$

There are no common factors to cancel.
The simplified expression is $\boxed{\frac{(2x - 1)(2x + 1)}{4x(x - 2)}}$.

It is a difference of squares.

$$a^2 - 25 = (a - 5)(a + 5)$$

Note that $5+a$ is the same as $a+5$.

$$\frac{(a - 5)(a + 5)}{(a-5)^2} \times \frac{a-5}{a+5}$$

$$\frac{(a - 5)(a + 5)}{(a-5)(a-5)} \times \frac{a-5}{a+5} = 1$$

The simplified expression is $\boxed{1}$.

$x^2 - 1$ is a difference of squares.

$$x^2 - 1 = (x - 1)(x + 1)$$

$$\frac{x + 1}{x - 1}$$

The simplified expression is $\boxed{\frac{x + 1}{x - 1}}$.