Question

1. (\frac{1}{a - 8} cdot \frac{a^2 - 3a - 18}{a + 3})

Explanation:

Step1: Factor the quadratic

Factor \(a^2 - 3a - 18\). We need two numbers that multiply to \(-18\) and add to \(-3\). Those numbers are \(-6\) and \(3\). So, \(a^2 - 3a - 18=(a - 6)(a + 3)\).
The expression becomes \(\frac{1}{a - 8}\cdot\frac{(a - 6)(a + 3)}{a + 3}\).

Step2: Cancel common factors

Cancel the common factor \((a + 3)\) in the numerator and denominator.
We are left with \(\frac{1}{a - 8}\cdot(a - 6)\).

Step3: Multiply the fractions

Multiply the remaining terms: \(\frac{a - 6}{a - 8}\).

Answer:

\(\frac{a - 6}{a - 8}\)