Questions: 6/(x^2-4) + x/(x+2) ([?] x^2+□ x+□)/((x-2)(x+□))

Solution Steps

To simplify the given expression, we need to find a common denominator for the fractions involved. The first fraction has a denominator of \(x^2 - 4\), which can be factored into \((x-2)(x+2)\). The second fraction already has a denominator of \(x+2\). Once we have a common denominator, we can combine the fractions and simplify the resulting expression.

We start with the two fractions: \[ \frac{6}{x^2 - 4} + \frac{x}{x + 2} \] The first fraction can be factored as: \[ x^2 - 4 = (x - 2)(x + 2) \] Thus, we rewrite the expression as: \[ \frac{6}{(x - 2)(x + 2)} + \frac{x}{x + 2} \]

The common denominator for the two fractions is \((x - 2)(x + 2)\). We rewrite the second fraction: \[ \frac{x}{x + 2} = \frac{x(x - 2)}{(x - 2)(x + 2)} \] Now, we can combine the fractions: \[ \frac{6 + x(x - 2)}{(x - 2)(x + 2)} \]

Expanding the numerator: \[ x(x - 2) = x^2 - 2x \] Thus, the combined expression becomes: \[ \frac{6 + x^2 - 2x}{(x - 2)(x + 2)} = \frac{x^2 - 2x + 6}{(x - 2)(x + 2)} \]

Final Answer