Question

enter the correct answer in the box. what is the simplest form of the expression representing this product? \\(\frac{x + 10}{x^2 + 7x - 18} \cdot \frac{3x^2 - 12x + 12}{3x + 30}\\)

Explanation:

Step1: Factor each expression

First, factor the quadratic in the denominator of the first fraction: \(x^2 + 7x - 18\). We need two numbers that multiply to \(-18\) and add to \(7\). Those numbers are \(9\) and \(-2\), so \(x^2 + 7x - 18=(x + 9)(x - 2)\).

Next, factor the numerator of the second fraction: \(3x^2 - 12x + 12\). First, factor out the greatest common factor, which is \(3\): \(3(x^2 - 4x + 4)\). Then, \(x^2 - 4x + 4\) is a perfect square trinomial, so \(x^2 - 4x + 4=(x - 2)^2\). Thus, \(3x^2 - 12x + 12 = 3(x - 2)^2\).

Also, factor the denominator of the second fraction: \(3x + 30\). Factor out the greatest common factor, which is \(3\): \(3(x + 10)\).

So now the expression becomes: \(\frac{x + 10}{(x + 9)(x - 2)} \cdot \frac{3(x - 2)^2}{3(x + 10)}\)

Step2: Cancel out common factors

We can cancel out the common factors in the numerator and the denominator. The \(x + 10\) terms cancel out, the \(3\) terms cancel out, and one \((x - 2)\) term cancels out (since we have \((x - 2)^2\) in the numerator and \((x - 2)\) in the denominator).

After canceling, we are left with \(\frac{x - 2}{x + 9}\)

Answer:

\(\frac{x - 2}{x + 9}\)