Question

what is the least common denominator of the expression below?
\\(\frac{x^2 + 6}{x^2 + 4x - 12} + \frac{7x}{x^2 + 7x + 6}\\)
a. \\(x^2 + 4x + 4\\)
b. \\((x + 6)\\)
c. \\((x + 6)(x - 2)(x + 1)\\)
d. \\((x + 6)(x - 2)(x + 6)(x + 1)\\)

Explanation:

Step1: Factor the first denominator

Factor \(x^2 + 4x - 12\). We need two numbers that multiply to \(-12\) and add to \(4\). Those numbers are \(6\) and \(-2\). So, \(x^2 + 4x - 12=(x + 6)(x - 2)\).

Step2: Factor the second denominator

Factor \(x^2 + 7x + 6\). We need two numbers that multiply to \(6\) and add to \(7\). Those numbers are \(6\) and \(1\). So, \(x^2 + 7x + 6=(x + 6)(x + 1)\).

Step3: Find the least common denominator (LCD)

The LCD of two fractions is the least common multiple of their denominators. For the denominators \((x + 6)(x - 2)\) and \((x + 6)(x + 1)\), we take the product of the highest power of each distinct factor. The distinct factors are \((x + 6)\), \((x - 2)\), and \((x + 1)\). So the LCD is \((x + 6)(x - 2)(x + 1)\).

Answer:

C. \((x + 6)(x - 2)(x + 1)\)