Simplify the following as a single fraction in its simplest form. (a) \( \frac{x^{2}-5 x-6}{3 x+3} \times \frac{x-2}{x^{2}-5 x+6} \)

To simplify the expression \( \frac{x^{2}-5 x-6}{3 x+3} \times \frac{x-2}{x^{2}-5 x+6} \), we first need to factor the polynomials in the numerators and the denominators. 1. Factor \(x^{2}-5x-6\): \[ x^{2}-5x-6 = (x-6)(x+1) \] 2. Factor \(3x+3\): \[ 3x+3 = 3(x+1) \] 3. Factor \(x^{2}-5x+6\): \[ x^{2}-5x+6 = (x-2)(x-3) \] Now, we can rewrite the expression: \[ \frac{(x-6)(x+1)}{3(x+1)} \times \frac{x-2}{(x-2)(x-3)} \] Now, cancel common factors: - The \( (x+1) \) terms can be canceled from the numerator and denominator. - \( (x-2) \) also cancels from the numerator and denominator. The expression now simplifies to: \[ \frac{x-6}{3(x-3)} \] Thus, the final simplified form of the expression is: \[ \frac{x-6}{3(x-3)} \]