Final answer:
The expression (x²+5x+6)/(x²-6x+9) * (5x-15)/(x²-9) simplifies to 5(x+2)/(x-3) after factoring out common terms in the numerator and denominator and then canceling them out.
Step-by-step explanation:
To multiply and simplify the expression (x²+5x+6)/(x²-6x+9) * (5x-15)/(x²-9), we first look for factors that can cancel out to simplify the expression.
The first step is to factor the quadratics where possible. The numerator x²+5x+6 can be factored into (x+2)(x+3). The denominator x²-6x+9 is a perfect square and can be factored into (x-3)(x-3) or (x-3)². The second numerator 5x-15 can be factored out to 5(x-3), and the second denominator x²-9 is a difference of squares, which factors into (x+3)(x-3).
Thus the expression becomes:
(x+2)(x+3) / (x-3)² * 5(x-3) / (x+3)(x-3)
We can cancel out the common terms (x+3) and (x-3) present in both the numerator and the denominator, leading to further simplification:
(x+2) / (x-3) * 5
Multiplying the remaining terms gives us the final simplified expression:
5(x+2)/(x-3)
This simplified result is our answer.