【Solution】Solution: Grouping (4 and 5 in one group, the rest are in their own groups.)
First case 4 and 5 are the same color, the number of ways to choose is: $6 \times 3 \times 3 \times 3 \times 3=324$ (ways).
Second case 4 and 5 are different colors, the number of ways to paint is: $12 \times 2 \times 2 \times 3 \times 3=432$ (ways).
$$
324+432=756 \text { (ways) }
$$

Therefore: There are a total of 756 different ways to paint.
8. (10 points) Use 4 different colors to color the circles (not all 4 colors need to be used). The requirement is that two circles directly connected by a line must have different colors. Then there are $\qquad$ different coloring methods.