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) ]

Multiply and simplify. [ (x^2-7 x-18)/(2 x^2-162) cdot (x^2+18 x+81)/(2 x^2+22 x+36) ]
Transcript text: Multiply and simplify. \[ \frac{x^{2}-7 x-18}{2 x^{2}-162} \cdot \frac{x^{2}+18 x+81}{2 x^{2}+22 x+36} \]
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Solution

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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.
Step 1: Factorization

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 \( 2(x + 2)(x + 9) \).
Step 2: Simplification of Fractions

We can now express the fractions using their factored forms:

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

Step 3: Canceling Common Factors

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

Thus, the simplified result of the multiplication is:

\[ \boxed{\frac{1}{4}} \]

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