Question

consider the expression below.
\\(\frac{3x + 6}{x^2 - x - 6} + \frac{2x}{x^2 + x - 12}\\)
place the steps required to determine the sum of the two expressions in the correct order.
\\(\frac{3(x + 2)}{(x + 2)(x - 3)} + \frac{2x}{(x - 3)(x + 4)}\\)
\\(\frac{5x + 12}{(x - 3)(x + 4)}\\)
\\(\frac{3}{(x - 3)} + \frac{2x}{(x - 3)(x + 4)}\\)
\\(\frac{3(x + 4)}{(x - 3)(x + 4)} + \frac{2x}{(x - 3)(x + 4)}\\)
\\(\frac{(3x + 12) + 2x}{(x - 3)(x + 4)}\\)
\\(\frac{5x + 6}{x^2 - x - 12}\\)
\\(\frac{(3x + 12) + 2x}{(x - 3)(x + 4)}\\)

Explanation:

Step1: Factor Denominators

Factor \(x^2 - x - 6=(x + 2)(x - 3)\) and \(x^2 + x - 12=(x - 3)(x + 4)\), also factor numerator \(3x + 6 = 3(x + 2)\). So we get \(\frac{3(x + 2)}{(x + 2)(x - 3)}+\frac{2x}{(x - 3)(x + 4)}\).

Step2: Simplify First Fraction

Cancel \((x + 2)\) in the first fraction: \(\frac{3}{(x - 3)}+\frac{2x}{(x - 3)(x + 4)}\).

Step3: Find Common Denominator

The common denominator of \(\frac{3}{x - 3}\) and \(\frac{2x}{(x - 3)(x + 4)}\) is \((x - 3)(x + 4)\). Rewrite \(\frac{3}{x - 3}\) as \(\frac{3(x + 4)}{(x - 3)(x + 4)}\), so we have \(\frac{3(x + 4)}{(x - 3)(x + 4)}+\frac{2x}{(x - 3)(x + 4)}\).

Step4: Add Numerators

Add the numerators: \(\frac{3(x + 4)+2x}{(x - 3)(x + 4)}=\frac{3x + 12+2x}{(x - 3)(x + 4)}=\frac{5x + 12}{(x - 3)(x + 4)}\) (Note: There might be a typo in the given options, but following the correct addition, we proceed with the steps as per factoring and adding).

Answer:

1. \(\frac{3(x + 2)}{(x + 2)(x - 3)}+\frac{2x}{(x - 3)(x + 4)}\)
2. \(\frac{3}{(x - 3)}+\frac{2x}{(x - 3)(x + 4)}\)
3. \(\frac{3(x + 4)}{(x - 3)(x + 4)}+\frac{2x}{(x - 3)(x + 4)}\)
4. \(\frac{(3x + 12)+2x}{(x - 3)(x + 4)}\) (or \(\frac{5x + 12}{(x - 3)(x + 4)}\) after combining like terms)