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What is the simplest form of the expression representing this product?

[tex] \frac{x+10}{x^2+7x-18} \cdot \frac{3x^2-12x+12}{3x+30} [/tex]

[tex] \boxed{\phantom{answer}} [/tex]



Answer :

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]