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\left(5m^{50}-11n^{8}\right)\left(5m^{50}+11n^{8}\right)
Rewrite 25m^{100}-121n^{16} as \left(5m^{50}\right)^{2}-\left(11n^{8}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).