Add. 29. \( \frac{4 x}{x+4}+\frac{6}{x+2} \)

To add the two rational expressions: \[ \frac{4x}{x + 4} + \frac{6}{x + 2} \] follow these steps: ### Step 1: Find a Common Denominator The denominators are \( x + 4 \) and \( x + 2 \). The **Least Common Denominator (LCD)** is the product of these two since they are distinct linear factors: \[ \text{LCD} = (x + 4)(x + 2) \] ### Step 2: Rewrite Each Fraction with the LCD Adjust each fraction so that both have the LCD as their denominator: 1. **First Fraction:** \[ \frac{4x}{x + 4} = \frac{4x \cdot (x + 2)}{(x + 4)(x + 2)} = \frac{4x(x + 2)}{(x + 4)(x + 2)} \] 2. **Second Fraction:** \[ \frac{6}{x + 2} = \frac{6 \cdot (x + 4)}{(x + 4)(x + 2)} = \frac{6(x + 4)}{(x + 4)(x + 2)} \] ### Step 3: Add the Numerators Now that both fractions have the same denominator, you can combine them: \[ \frac{4x(x + 2) + 6(x + 4)}{(x + 4)(x + 2)} \] Expand and simplify the numerator: \[ 4x(x + 2) + 6(x + 4) = 4x^2 + 8x + 6x + 24 = 4x^2 + 14x + 24 \] ### Step 4: Simplify the Combined Fraction The expression now is: \[ \frac{4x^2 + 14x + 24}{(x + 4)(x + 2)} \] You can factor out a common factor in the numerator if possible. Let's check: \[ 4x^2 + 14x + 24 = 2(2x^2 + 7x + 12) \] Attempting to factor \(2x^2 + 7x + 12\): We look for two numbers that multiply to \(2 \times 12 = 24\) and add to \(7\). Unfortunately, there are no integer pairs that satisfy this condition. Therefore, the numerator cannot be factored further. ### Final Answer The sum of the two rational expressions is: \[ \frac{4x^2 + 14x + 24}{(x + 4)(x + 2)} \] Or, factoring out a common factor of 2: \[ \frac{2(2x^2 + 7x + 12)}{(x + 4)(x + 2)} \] This is the simplest form of the sum of the given expressions.