Questions: Multiply and simplify the following rational expression. (x^2+12x+35)/(x^2+3x-28) * (x-4)/(x+1) Give your answer as a reduced rational expression.

Transcript text: Multiply and simplify the following rational expression. \[ \frac{x^{2}+12 x+35}{x^{2}+3 x-28} \cdot \frac{x-4}{x+1} \] Give your answer as a reduced rational expression.

Solution

Solution Steps

To multiply and simplify the given rational expression, follow these steps:

Factorize the numerators and denominators of both rational expressions.
Multiply the factored forms.
Cancel out any common factors in the numerator and denominator.
Simplify the resulting expression.

Step 1: Factorize the Numerators and Denominators

First, we factorize the numerators and denominators of the given rational expressions: \[ \frac{x^2 + 12x + 35}{x^2 + 3x - 28} \cdot \frac{x - 4}{x + 1} \]

The factorizations are: \[ x^2 + 12x + 35 = (x + 5)(x + 7) \] \[ x^2 + 3x - 28 = (x + 7)(x - 4) \]

So, the rational expressions become: \[ \frac{(x + 5)(x + 7)}{(x + 7)(x - 4)} \cdot \frac{x - 4}{x + 1} \]

Step 2: Multiply the Factored Forms

Next, we multiply the factored forms: \[ \frac{(x + 5)(x + 7)}{(x + 7)(x - 4)} \cdot \frac{x - 4}{x + 1} = \frac{(x + 5)(x + 7)(x - 4)}{(x + 7)(x - 4)(x + 1)} \]

Step 3: Cancel Out Common Factors

We cancel out the common factors \((x + 7)\) and \((x - 4)\) from the numerator and the denominator: \[ \frac{(x + 5) \cancel{(x + 7)} \cancel{(x - 4)}}{\cancel{(x + 7)} \cancel{(x - 4)} (x + 1)} = \frac{x + 5}{x + 1} \]