b. \( \frac{x^{2}+7 x-18}{x^{2}-5 x+6} \)

To tackle the expression \( \frac{x^{2}+7 x-18}{x^{2}-5 x+6} \), let's first factor both the numerator and the denominator. The numerator \( x^{2} + 7x - 18 \) can be factored into \( (x + 9)(x - 2) \), while the denominator \( x^{2} - 5x + 6 \) factors into \( (x - 2)(x - 3) \). So, the expression can be rewritten as: \[ \frac{(x + 9)(x - 2)}{(x - 2)(x - 3)} \]. We can simplify this by canceling the common factor \( (x - 2) \), leading us to the simpler expression: \[ \frac{x + 9}{x - 3} \] (for \( x \neq 2 \)). This simplification allows for easier evaluation of the expression at various values of \( x \). Now, let’s see what happens with specific values of \( x \) and common pitfalls to avoid! When evaluating \( x = 4 \) in the simplified expression \( \frac{x + 9}{x - 3} \), we get \( \frac{4 + 9}{4 - 3} = \frac{13}{1} = 13 \). However, if you substitute \( x = 2 \) in the original expression, you’ll run into a problem, since it leads to division by zero in the denominator! Always make sure to check for any restrictions from the original expression after simplifications. Understanding these factors can help you avoid mistakes and improve your problem-solving skills in algebra!