To simplify the expression:

$\frac{x^2 - 7x + 12}{x^2 - x - 2x + 1}$

given that $x \neq 4$ and $x \neq -1$, we will follow these steps:

1. Factorize the Numerator and Denominator:
   We start by factorizing both the numerator and the denominator.

   • Numerator: $x^2 - 7x + 12$

     • Find two numbers that multiply to 12 and add up to -7, which are -3 and -4.
     • Thus, $x^2 - 7x + 12 = (x - 3)(x - 4)$.

   • Denominator: $x^2 - x - 2x + 1$

     • Group terms: $x^2 - 3x + 1$
     • Further simplify