Consider the following transformation:
$$11^8 + 11^7 - 132 = 11^6 \cdot (11^2 + 11) - 132 = 11^6 \cdot 132 - 132 = 132 \cdot (11^6 - 1).$$
Since $132 = 11 \cdot 12 = 2^2 \cdot 3 \cdot 11$, it has three prime divisors: $2$, $3$, $11$.
$$\begin{aligned} 11^6 - 1 &= (11^3 - 1)(11^3 + 1) = (11-1)(11^2 + 11+1)(11+1)(11^2 - 11+1) \\ &= 10 \cdot 133 \cdot 12 \cdot 111 = 2^3 \cdot 3^2 \cdot 5 \cdot 7 \cdot 19 \cdot 37. \end{aligned}$$
Therefore, we have four more prime divisors: $5$, $7$, $19$, $37$.