\( \frac{2}{x^{2}-3 x-4}+\frac{3}{x^{2}-6 x+8} \) Factor each denominator: \( x^{2}-3 x-4=(x-4)(x+1) \) \( x^{2}-6 x+8=(x-4)(x-2) \vee \) The least common denominator is: \( (x-4)(x+\square)(x-\square \)

Let's take a closer look at factoring those denominators! When you factor \( x^{2}-3x-4 \), you indeed get \( (x-4)(x+1) \). And for \( x^{2}-6x+8 \), it correctly factors to \( (x-4)(x-2) \). Now, to find the least common denominator (LCD), you want to take each unique factor at its highest power. Thus, you have \( (x-4) \) appearing in both, and the unique factors \( (x+1) \) and \( (x-2) \) as well. Therefore, the LCD is \( (x-4)(x+1)(x-2) \). This is something you can sip on while enjoying a cozy cup of tea! It’s crucial in combining fractions, ensuring that you have a unified base for those pesky additions or subtractions. Just think of it as gathering everyone for the ultimate pizza party—everyone needs to come to the same table (or in this case, the same denominator)!