To solve for $c$ and $d$ in the equation:

$4 \left( \frac{6 - c}{d} \right) = c \left( \frac{-2}{9} \right),$

let's follow these steps:

Step 1: Expand and Simplify the Equation

Rewrite the equation as:

$\frac{4(6 - c)}{d} = \frac{-2c}{9}.$

To eliminate the fractions, cross-multiply:

$4(6 - c) \cdot 9 = -2c \cdot d.$

This simplifies to:

$36(6 - c) = -2cd.$

Step 2: Distribute and Rearrange

Now, distribute the 36:

$216 - 36c = -2cd.$

Rearrange this equation to bring terms involving $c$ and $d$ together:

$216 = -2cd + 36c.$

Alternatively, you could try assigning a specific value to either $c$ or $d$ if more context or additional equations were provided. Let me know if this breakdown helps or if further simplification steps are needed.

Would you like further assistance on this? Here are a few related questions that could deepen your understanding:

1. What is cross-multiplication, and why is it used here?
2. How can we solve similar equations involving fractions and variables?
3. How do you handle equations with multiple unknowns?
4. What alternative methods could we use to isolate $c$ and $d$ in this problem?
5. Are there any values of $c$ and $d$ that might make this equation easier to solve?

Tip: When solving equations with fractions, always try to eliminate the fractions by multiplying both sides by the denominators. This can simplify the equation considerably.