Question 1210161
Let's solve this problem step-by-step.

**1. Treat Lancelot and Gawain as a Single Unit**

Since Sir Lancelot and Sir Gawain insist on sitting next to each other, we can treat them as a single unit. This unit can be arranged in two ways: Lancelot-Gawain or Gawain-Lancelot.

**2. Arrange the Unit and the Remaining Knights**

Now, we have 7 entities to arrange around the circular table: the Lancelot-Gawain unit and the remaining 6 knights. The number of ways to arrange $n$ distinct objects in a circle is $(n-1)!$.

Thus, we have $(7-1)! = 6!$ ways to arrange these 7 entities.

**3. Account for the Arrangement of Lancelot and Gawain**

Since Lancelot and Gawain can switch places within their unit, we need to multiply the number of arrangements by 2.

**4. Calculate the Total Number of Arrangements**

Total number of arrangements = (arrangements of the unit and other knights) * (arrangements of Lancelot and Gawain)

Total number of arrangements = $6! \times 2$

$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$

Total number of arrangements = $720 \times 2 = 1440$

**Therefore, there are 1440 ways for the 8 knights to be seated if Sir Lancelot and Sir Gawain insist on sitting next to each other.**