Questions: Multiply and simplify. [ (x^2-7 x-18)/(2 x^2-162) cdot (x^2+18 x+81)/(2 x^2+22 x+36) ]

Solution Steps

To solve the given problem, we need to follow these steps:

1. Factorize the numerators and denominators of both fractions.
2. Simplify the fractions by canceling out common factors.
3. Multiply the simplified fractions.

We start by factorizing the expressions in the problem:

1. The numerator $x^2 - 7x - 18$ factors to $(x - 9)(x + 2)$.
2. The denominator $2x^2 - 162$ factors to $2(x - 9)(x + 9)$.
3. The numerator $x^2 + 18x + 81$ factors to $(x + 9)^2$.
4. The denominator $2x^2 + 22x + 36$ factors to $2(x + 2)(x + 9)$.

We can now express the fractions using their factored forms:

$$\frac{(x - 9)(x + 2)}{2(x - 9)(x + 9)} \cdot \frac{(x + 9)^2}{2(x + 2)(x + 9)}$$

Next, we simplify the expression by canceling out the common factors:

• The $(x - 9)$ in the numerator and denominator cancels.
• The $(x + 2)$ in the numerator and denominator cancels.
• One $(x + 9)$ in the numerator cancels with one in the denominator.

This leaves us with:

$$\frac{(x + 9)}{4}$$

Final Answer