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.

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.
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Solution

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Solution Steps

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

  1. Factorize the numerators and denominators of both rational expressions.
  2. Multiply the factored forms.
  3. Cancel out any common factors in the numerator and denominator.
  4. 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: \[ \frac{(x + 5) \cancel{(x + 7)} \cancel{(x - 4)}}{\cancel{(x + 7)} \cancel{(x - 4)} (x + 1)} = \frac{x + 5}{x + 1} \]

Final Answer

The simplified rational expression is: \[ \boxed{\frac{x + 5}{x + 1}} \]

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