Question

complete the following steps to find the lcd and write the sum of the numerators for the given problem:
\\(\frac{2}{x^2 - 3x - 4} + \frac{3}{x^2 - 6x + 8}\\)
factor each denominator:
\\(x^2 - 3x - 4 = (x - 4)(x + 1)\\)
\\(x^2 - 6x + 8 = \\)
options:
\\((x + 8)(x - 6)\\)
\\((x - 4)(x + 1)\\)
\\((x - 4)(x - 2)\\)

Explanation:

Step1: Factor \(x^2 - 6x + 8\)

We need two numbers that multiply to \(8\) and add to \(-6\). The numbers are \(-4\) and \(-2\) because \((-4)\times(-2)=8\) and \((-4)+(-2)=-6\). So, we can factor the quadratic as \(x^2 - 6x + 8=(x - 4)(x - 2)\).

Step2: Verify the factors

Multiply \((x - 4)(x - 2)\) using the distributive property (FOIL method):
First: \(x\times x = x^2\)
Outer: \(x\times(-2)=-2x\)
Inner: \(-4\times x=-4x\)
Last: \(-4\times(-2)=8\)
Combine like terms: \(x^2-2x - 4x + 8=x^2-6x + 8\), which matches the original quadratic.

Answer:

\((x - 4)(x - 2)\)