Question

(x^(2)+8x+12)/(x^(2)+10x+24)*(x^(2)+4x)/(x^(2)-3x-10) perform the indicated operation on rational expression simplify if possible

Answer 1

To simplify the given rational expression, we need to factor the quadratic polynomials in the numerators and denominators, and then cancel out any common factors. The simplified form of the expression is x / (x - 5).

Step-by-step explanation:

To simplify the given rational expression, we must factor the numerators and denominators where possible and cancel out any common factors. Let's break down the expression step by step:

The original expression is:

(x² + 8x + 12) / (x² + 10x + 24) * (x² + 4x) / (x² - 3x - 10)

First, we need to factor each quadratic polynomial:

• The numerator of the first fraction, x² + 8x + 12 can be factored into (x + 2)(x + 6)
• The denominator of the first fraction, x² + 10x + 24 can be factored into (x + 4)(x + 6)
• The numerator of the second fraction, x² + 4x, can be factored by taking out a common factor x, resulting in x(x + 4)
• The denominator of the second fraction, x² - 3x - 10, can be factored into (x - 5)(x + 2)

Substituting these factors back into the original expression, we have:

((x + 2)(x + 6)) / ((x + 4)(x + 6)) * (x(x + 4)) / ((x - 5)(x + 2))

Now cancel out any common factors in the numerator and denominator:

• (x + 6) cancels out
• (x + 2) cancels out
• (x + 4) cancels out

After cancelling, the simplified expression is:

x / (x - 5)

This is the simplified form of the original expression.