To factor the quadratic expression \( n^2 - 12n + 11 \), we need to find two numbers that multiply to the constant term (11) and add up to the coefficient of the linear term (-12).
The two numbers that satisfy these conditions are -1 and -11, because:
\[ -1 \times -11 = 11 \quad \text{(product)} \] \[ -1 + (-11) = -12 \quad \text{(sum)} \]
Thus, we can factor the expression as follows:
\[ n^2 - 12n + 11 = (n - 1)(n - 11) \]
So the factored form of \( n^2 - 12n + 11 \) is:
\[ (n - 1)(n - 11) \]