To solve these algebraic expressions, we need to simplify each part step by step.

### Part (a)
\[ \frac{21}{x-3} + \frac{9-6x}{x^2-3x} \]

First, factor the denominator of the second fraction:
\[ x^2 - 3x = x(x-3) \]

So the expression becomes:
\[ \frac{21}{x-3} + \frac{9-6x}{x(x-3)} \]

To combine these fractions, find a common denominator, which is \( x(x-3) \):
\[ \frac{21 \cdot x}{(x-3) \cdot x} + \frac{9-6x}{x(x-3)} \]
\[ \frac{21x}{x(x-3)} + \frac{9-6x}{x(x-3)} \]

Combine the numerators:
\[ \frac{21x + 9 - 6x}{x(x-3)} \]
\[ \frac{15x + 9}{x(x-3)} \]

So, the simplified form of part (a) is:
\[ \frac{15x + 9}{x(x-3)} \]

### Part (b)
\[ \frac{4}{x+2} + \frac{2}{x-2} + \frac{5x-6}{x^2-4} \]

First, factor the denominator of the third fraction:
\[ x^2 - 4 = (x+2)(x-2) \]

So the expression becomes:
\[ \frac{4}{x+2} + \frac{2}{x-2} + \frac{5x-6}{(x+2)(x-2)} \]

To combine these fractions, find a common denominator, which is \( (x+2)(x-2) \):
\[ \frac{4 \cdot (x-2)}{(x+2)(x-2)} + \frac{2 \cdot (x+2)}{(x+2)(x-2)} + \frac{5x-6}{(x+2)(x-2)} \]
\[ \frac{4(x-2) + 2(x+2) + 5x-6}{(x+2)(x-2)} \]

Simplify the numerator:
\[ 4(x-2) + 2(x+2) + 5x-6 \]
\[ 4x - 8 + 2x + 4 + 5x - 6 \]
\[ (4x + 2x + 5x) + (-8 + 4 - 6) \]
\[ 11x - 10 \]

So, the simplified form of part (b) is:
\[ \frac{11x - 10}{(x+2)(x-2)} \]

Therefore, the simplified forms are:
(a) \[ \frac{15x + 9}{x(x-3)} \]
(b) \[ \frac{11x - 10}{(x+2)(x-2)} \]