Answer :
To solve the given problem, we need to simplify the expression:
[tex]\[ \frac{x^2 - x - 6}{x^2 + 6x - 27} \cdot \frac{x^2 + x - 72}{x^2 + 11x + 18} \][/tex]
### Step 1: Factorize Each Polynomial
First, we factorize each polynomial in the numerators and denominators.
- For the numerator \( x^2 - x - 6 \):
[tex]\[ x^2 - x - 6 = (x - 3)(x + 2) \][/tex]
- For the denominator \( x^2 + 6x - 27 \):
[tex]\[ x^2 + 6x - 27 = (x - 3)(x + 9) \][/tex]
- For the numerator \( x^2 + x - 72 \):
[tex]\[ x^2 + x - 72 = (x - 8)(x + 9) \][/tex]
- For the denominator \( x^2 + 11x + 18 \):
[tex]\[ x^2 + 11x + 18 = (x + 2)(x + 9) \][/tex]
### Step 2: Substitute the Factored Forms
Substitute the factored forms back into the expression:
[tex]\[ \frac{(x - 3)(x + 2)}{(x - 3)(x + 9)} \cdot \frac{(x - 8)(x + 9)}{(x + 2)(x + 9)} \][/tex]
### Step 3: Simplify the Expression
Next, we can simplify by canceling out common factors in the numerator and the denominator.
- The factor \((x - 3)\) occurs in both the numerator and the denominator of the first fraction and can be canceled.
- The factor \((x + 2)\) occurs in both the numerator of the first fraction and the denominator of the second fraction and can be canceled.
- The factor \((x + 9)\) occurs in both the numerator of the second fraction and the denominator of both the first and second fractions and can be canceled.
After canceling these factors, we are left with:
[tex]\[ \frac{(x - 8)}{(x + 9)} \][/tex]
### Final Result
The simplified expression, in factored form, is:
[tex]\[ \boxed{\frac{x - 8}{x + 9}} \][/tex]
[tex]\[ \frac{x^2 - x - 6}{x^2 + 6x - 27} \cdot \frac{x^2 + x - 72}{x^2 + 11x + 18} \][/tex]
### Step 1: Factorize Each Polynomial
First, we factorize each polynomial in the numerators and denominators.
- For the numerator \( x^2 - x - 6 \):
[tex]\[ x^2 - x - 6 = (x - 3)(x + 2) \][/tex]
- For the denominator \( x^2 + 6x - 27 \):
[tex]\[ x^2 + 6x - 27 = (x - 3)(x + 9) \][/tex]
- For the numerator \( x^2 + x - 72 \):
[tex]\[ x^2 + x - 72 = (x - 8)(x + 9) \][/tex]
- For the denominator \( x^2 + 11x + 18 \):
[tex]\[ x^2 + 11x + 18 = (x + 2)(x + 9) \][/tex]
### Step 2: Substitute the Factored Forms
Substitute the factored forms back into the expression:
[tex]\[ \frac{(x - 3)(x + 2)}{(x - 3)(x + 9)} \cdot \frac{(x - 8)(x + 9)}{(x + 2)(x + 9)} \][/tex]
### Step 3: Simplify the Expression
Next, we can simplify by canceling out common factors in the numerator and the denominator.
- The factor \((x - 3)\) occurs in both the numerator and the denominator of the first fraction and can be canceled.
- The factor \((x + 2)\) occurs in both the numerator of the first fraction and the denominator of the second fraction and can be canceled.
- The factor \((x + 9)\) occurs in both the numerator of the second fraction and the denominator of both the first and second fractions and can be canceled.
After canceling these factors, we are left with:
[tex]\[ \frac{(x - 8)}{(x + 9)} \][/tex]
### Final Result
The simplified expression, in factored form, is:
[tex]\[ \boxed{\frac{x - 8}{x + 9}} \][/tex]
