Well, let's see... We have 5 boys and 3 girls, so there are a total of 8 people. If we ignore the restriction about Alex and Beth, there would be 8! (factorial) possible arrangements. However, we need to eliminate the arrangements where Alex and Beth are seated next to each other.
Now, think about it this way. If we treat Alex and Beth as a single entity, there would be 7 entities (6 single boys and the "AlexBeth" pair) to arrange around the table. There are 7! possible arrangements when Alex and Beth are seated next to each other.
But we don't want that. So, we need to consider the cases where Alex and Beth are NOT seated next to each other. There are two ways for that to happen - either Alex is on the left of Beth, or he is on the right. So, for each of these cases, there would be 7! arrangements.
Therefore, the total number of different arrangements where Alex and Beth are not seated next to each other will be 2 * 7!.
And if you want me to calculate the exact number, I'm afraid I don't have enough clown fingers to do the math for you. Maybe you could give it a try?