To determine the simplest form of the given product of expressions, let's break it down step-by-step.
Given:
[tex]\[ \left( \frac{x+10}{x^2 + 7x - 18} \right) \cdot \left( \frac{3x^2 - 12x + 12}{3x + 30} \right) \][/tex]
### Step 1: Factorize the Denominators and Numerators
First, let's factorize the polynomials in the denominators and numerators.
#### Numerator and Denominator of the First Expression:
[tex]\[ \frac{x+10}{x^2 + 7x - 18} \][/tex]
The denominator \(x^2 + 7x - 18\) can be factorized:
[tex]\[ x^2 + 7x - 18 = (x + 9)(x - 2) \][/tex]
So, the first expression becomes:
[tex]\[ \frac{x+10}{(x + 9)(x - 2)} \][/tex]
#### Numerator and Denominator of the Second Expression:
[tex]\[ \frac{3x^2 - 12x + 12}{3x + 30} \][/tex]
Factor out a common factor:
Numerator: \( 3x^2 - 12x + 12 = 3(x^2 - 4x + 4) = 3(x - 2)^2 \)
Denominator: \( 3x + 30 = 3(x + 10) \)
So, the second expression becomes:
[tex]\[ \frac{3(x - 2)^2}{3(x + 10)} \][/tex]
### Step 2: Simplify the Product of the Expressions
Now, combine the simplified forms of both expressions:
[tex]\[ \frac{x+10}{(x + 9)(x - 2)} \cdot \frac{3(x - 2)^2}{3(x + 10)} \][/tex]
Notice that \(3\) in the numerator and denominator will cancel out. Also, \(x+10\) in the numerator of the first fraction and the denominator of the second fraction will cancel out.
This gives us:
[tex]\[ \frac{x+10}{(x + 9)(x - 2)} \cdot \frac{(x - 2)^2}{(x + 10)} \][/tex]
Cancel out \(x + 10\) and simplify the remaining:
[tex]\[ \frac{(x-2)^2}{(x + 9)(x - 2)} \][/tex]
One \( (x - 2) \) in the numerator and denominator cancels out:
[tex]\[ \frac{x - 2}{x + 9} \][/tex]
We reach the simplest form:
[tex]\[ \boxed{\frac{x - 2}{x + 9}} \][/tex]
