Simplify the rational expressions below:
$\frac{x^2-4x}{x^2-2x-8x}$

• A. $\frac{x-4}{x-10}$
• B. $\frac{x+4}{x+10}$
• C. $\frac{x-2}{x-5}$
• D. $\frac{x+2}{x+5}$

Answer: (A)

Verified step by step guidance

1. Start by carefully examining the given rational expression: \(\frac{x^2 - 4x}{x^2 - 2x - 8x}\). Notice that the denominator can be simplified by combining like terms.
2. Combine the like terms in the denominator: \(x^2 - 2x - 8x\) becomes \(x^2 - 10x\). So the expression is now \(\frac{x^2 - 4x}{x^2 - 10x}\).
3. Next, factor both the numerator and the denominator. For the numerator \(x^2 - 4x\), factor out the common factor \(x\) to get \(x(x - 4)\). For the denominator \(x^2 - 10x\), factor out \(x\) to get \(x(x - 10)\).
4. Rewrite the expression using the factored forms: \(\frac{x(x - 4)}{x(x - 10)}\). Now, identify any common factors in the numerator and denominator that can be canceled.
5. Cancel the common factor \(x\) from numerator and denominator (noting that \(x \neq 0\) to avoid division by zero), leaving the simplified expression \(\frac{x - 4}{x - 10}\).