Questions: Simplify the expression (x^2-9)/(x-3) × (2x^2-2)/(x^2+4x+3)

Solution Steps

To simplify the given expression, we need to factorize the polynomials in the numerator and the denominator and then cancel out the common factors.

1. Factorize \(x^2 - 9\) as \((x - 3)(x + 3)\).
2. Factorize \(2x^2 - 2\) as \(2(x^2 - 1) = 2(x - 1)(x + 1)\).
3. Factorize \(x^2 + 4x + 3\) as \((x + 3)(x + 1)\).
4. Substitute these factorizations into the original expression.
5. Cancel out the common factors.

First, we factorize the polynomials in the numerator and the denominator.

\[ x^2 - 9 = (x - 3)(x + 3) \]

\[ 2x^2 - 2 = 2(x^2 - 1) = 2(x - 1)(x + 1) \]

\[ x^2 + 4x + 3 = (x + 3)(x + 1) \]

Next, we substitute these factorizations into the original expression:

\[ \frac{(x^2 - 9)(2x^2 - 2)}{(x - 3)(x^2 + 4x + 3)} = \frac{(x - 3)(x + 3) \cdot 2(x - 1)(x + 1)}{(x - 3)(x + 3)(x + 1)} \]

We then cancel out the common factors in the numerator and the denominator:

\[ \frac{(x - 3)(x + 3) \cdot 2(x - 1)(x + 1)}{(x - 3)(x + 3)(x + 1)} = 2(x - 1) \]

Final Answer