To factor out the greatest common factor (GCF) from the polynomial \(22n^{10} + 11n^5 + 11n^4 - 11n^2\), let's first identify the GCF of the coefficients and the variable terms.
- The coefficients are 22, 11, 11, and -11. The GCF of these numbers is 11.
- The variable terms have \(n^{10}\), \(n^5\), \(n^4\), and \(n^2\). The lowest power of \(n\) is \(n^2\).
Therefore, the GCF of the entire polynomial is \(11n^2\).
Now, we can factor out \(11n^2\):
\[ 22n^{10} + 11n^5 + 11n^4 - 11n^2 = 11n^2(2n^8 + n^3 + n^2 - 1) \]
So, the factored form of the polynomial is:
\[ \boxed{11n^2(2n^8 + n^3 + n^2 - 1)} \]