To simplify or combine the expression $\frac{4}{x+2} + \frac{3}{x+6}$, we need a common denominator. The denominators are $x+2$ and $x+6$, so the least common denominator (LCD) is $(x+2)(x+6)$.

Steps:

1. Rewrite each fraction with the LCD as the denominator: $\frac{4}{x+2} = \frac{4(x+6)}{(x+2)(x+6)}$ $\frac{3}{x+6} = \frac{3(x+2)}{(x+2)(x+6)}$

2. Combine the fractions: $\frac{4}{x+2} + \frac{3}{x+6} = \frac{4(x+6) + 3(x+2)}{(x+2)(x+6)}$

3. Expand the numerators: $4(x+6) = 4x + 24, \quad 3(x+2) = 3x + 6$ So, the numerator becomes: $4x + 24 + 3x + 6 = 7x + 30$

4. Write the final expression: $\frac{4}{x+2} + \frac{3}{x+6} = \frac{7x + 30}{(x+2)(x+6)}$

Final Answer:

$\frac{7x + 30}{(x+2)(x+6)}$

Would you like further explanation or to explore related examples?

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5 Related Questions:

1. How can this expression be used in solving equations?
2. What happens to the expression when $x = -2$ or $x = -6$?
3. How do you simplify similar fractions with quadratic denominators?
4. What are the steps to find the LCD for more than two fractions?
5. How does this relate to solving rational inequalities?

Tip:

When simplifying fractions, always check for excluded values (where denominators become zero) to ensure the expression is well-defined.