Sure! Let's solve the given expression step by step:
The given expression is:
[tex]\[
\frac{x^2 + x - 12}{x^2 + x - 72} \cdot \frac{x^2 + 7x + 12}{x^2 - x - 6} \div \frac{x + 4}{x^2 + 11x + 18}
\][/tex]
### Step 1: Factor all the polynomials
First, factor each polynomial where possible.
#### Numerators:
1. [tex]\( x^2 + x - 12 \)[/tex]
- This can be factored as [tex]\((x + 4)(x - 3)\)[/tex].
2. [tex]\( x^2 + 7x + 12 \)[/tex]
- This can be factored as [tex]\((x + 3)(x + 4)\)[/tex].
3. [tex]\( x + 4 \)[/tex]
- This is already a simplest form.
#### Denominators:
1. [tex]\( x^2 + x - 72 \)[/tex]
- This can be factored as [tex]\((x + 9)(x - 8)\)[/tex].
2. [tex]\( x^2 - x - 6 \)[/tex]
- This can be factored as [tex]\((x - 3)(x + 2)\)[/tex].
3. [tex]\( x^2 + 11x + 18 \)[/tex]
- This can be factored as [tex]\((x + 9)(x + 2)\)[/tex].
### Step 2: Substitute the factored forms
The expression now looks like:
[tex]\[
\frac{(x + 4)(x - 3)}{(x + 9)(x - 8)} \cdot \frac{(x + 3)(x + 4)}{(x - 3)(x + 2)} \div \frac{x + 4}{(x + 9)(x + 2)}
\][/tex]
### Step 3: Rewrite the division as multiplication by the reciprocal
[tex]\[
\frac{(x + 4)(x - 3)}{(x + 9)(x - 8)} \cdot \frac{(x + 3)(x + 4)}{(x - 3)(x + 2)} \cdot \frac{(x + 9)(x + 2)}{x + 4}
\][/tex]
### Step 4: Simplify by canceling common factors
1. Cancel [tex]\((x - 3)\)[/tex] in the numerator and denominator:
[tex]\[
\frac{(x + 4)}{(x + 9)(x - 8)} \cdot \frac{(x + 3)(x + 4)}{(x + 2)} \cdot \frac{(x + 9)(x + 2)}{x + 4}
\][/tex]
2. Cancel [tex]\((x + 4)\)[/tex] in the numerator and denominator:
[tex]\[
\frac{1}{(x + 9)(x - 8)} \cdot \frac{(x + 3)(x + 4)}{(x + 2)} \cdot \frac{(x + 9)(x + 2)}{1}
\][/tex]
3. Cancel [tex]\((x + 2)\)[/tex] in the numerator and denominator:
[tex]\[
\frac{1}{(x + 9)(x - 8)} \cdot (x + 3)(x + 4) \cdot (x + 9)
\][/tex]
4. Cancel [tex]\((x + 9)\)[/tex] in the numerator and denominator:
[tex]\[
\frac{1}{(x - 8)} \cdot (x + 3)(x + 4)
\][/tex]
The remaining expression is:
[tex]\[
\frac{(x + 3)(x + 4)}{x - 8}
\][/tex]
### Final Simplified Expression
[tex]\[
\frac{(x + 3)(x + 4)}{x - 8}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{(x + 3)(x + 4)}{x - 8}}
\][/tex]
