Let's simplify the given rational expression step-by-step.
1. Given expression:
[tex]\[
\frac{x^2-4x+3}{x^2+x-12}
\][/tex]
2. Factor the numerator and the denominator.
The numerator is [tex]\( x^2 - 4x + 3 \)[/tex].
To factor [tex]\( x^2 - 4x + 3 \)[/tex], we look for two numbers that multiply to 3 and add up to -4. These numbers are -1 and -3.
Therefore, [tex]\( x^2 - 4x + 3 = (x - 3)(x - 1) \)[/tex].
The denominator is [tex]\( x^2 + x - 12 \)[/tex].
To factor [tex]\( x^2 + x - 12 \)[/tex], we look for two numbers that multiply to -12 and add up to 1. These numbers are 4 and -3.
Therefore, [tex]\( x^2 + x - 12 = (x - 3)(x + 4) \)[/tex].
3. Rewrite the rational expression with the factored numerator and denominator:
[tex]\[
\frac{(x - 3)(x - 1)}{(x - 3)(x + 4)}
\][/tex]
4. Cancel out the common factor [tex]\((x - 3)\)[/tex] in both the numerator and the denominator:
[tex]\[
\frac{(x - 3)(x - 1)}{(x - 3)(x + 4)} = \frac{x - 1}{x + 4} \quad \text{for} \quad x \neq 3
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{x - 1}{x + 4}
\][/tex]
5. Comparing this result with the given choices, we find that:
B. [tex]\(\frac{x-1}{x+4}\)[/tex] is the correct answer.
Hence, the correct choice is B.
