Questions: Multiply the following and simplify completely: (x^2+7x-30)/(x^2+15x+50) * (x^2-25)/(x^2-9) Give your answer as a reduced rational expression. You may assume that the denominator is never 0.

Solution Steps

We start with the expression:

\[ \frac{x^{2}+7x-30}{x^{2}+15x+50} \cdot \frac{x^{2}-25}{x^{2}-9} \]

First, we factor each polynomial:

• The numerator \(x^{2}+7x-30\) factors to \((x - 5)(x + 6)\).
• The denominator \(x^{2}+15x+50\) factors to \((x + 5)(x + 10)\).
• The numerator \(x^{2}-25\) factors to \((x - 5)(x + 5)\).
• The denominator \(x^{2}-9\) factors to \((x - 3)(x + 3)\).

Thus, we have:

\[ \frac{(x - 5)(x + 6)}{(x + 5)(x + 10)} \cdot \frac{(x - 5)(x + 5)}{(x - 3)(x + 3)} \]

Next, we combine the two fractions:

\[ \frac{(x - 5)(x + 6)(x - 5)(x + 5)}{(x + 5)(x + 10)(x - 3)(x + 3)} \]

Now, we simplify the expression by canceling common factors in the numerator and denominator:

• The factor \((x - 5)\) appears twice in the numerator and once in the denominator.
• The factor \((x + 5)\) appears once in both the numerator and denominator.

After canceling, we are left with:

\[ \frac{(x - 5)}{(x + 3)} \]

Final Answer