1Cademy - Adding $$\frac{2n}{n^2-3n-10} + \frac{6}{n^2+5n+6}$$

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Adding $\frac{8}{x^2-2x-3} + \frac{3x}{x^2+4x+3}$

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Adding $\frac{2n}{n^2-3n-10} + \frac{6}{n^2+5n+6}$

Add two rational expressions with factorable trinomial denominators:

$\frac{2n}{n^2 - 3n - 10} + \frac{6}{n^2 + 5n + 6}$

Step 1 — Find the LCD and rewrite each fraction. Factor the denominators: $n^2 - 3n - 10 = (n - 5)(n + 2)$ and $n^2 + 5n + 6 = (n + 3)(n + 2)$ . The LCD is $(n - 5)(n + 2)(n + 3)$ . Multiply each fraction by its missing factor to rewrite them with the LCD:

$\frac{2n(n + 3)}{(n - 5)(n + 2)(n + 3)} + \frac{6(n - 5)}{(n + 3)(n + 2)(n - 5)}$

Distribute in each numerator: $2n(n + 3) = 2n^2 + 6n$ and $6(n - 5) = 6n - 30$ .

Step 2 — Add the numerators over the common denominator. Combine the numerators and collect like terms:

$\frac{2n^2 + 6n + 6n - 30}{(n - 5)(n + 2)(n + 3)} = \frac{2n^2 + 12n - 30}{(n - 5)(n + 2)(n + 3)}$

Step 3 — Simplify, if possible. Factor the numerator by first extracting the greatest common factor: $2n^2 + 12n - 30 = 2(n^2 + 6n - 15)$ . The trinomial $n^2 + 6n - 15$ is prime, as there are no two integers that multiply to $-15$ and add to $6$ . Because the factored numerator shares no common factors with the denominator, the expression cannot be simplified further:

$\frac{2(n^2 + 6n - 15)}{(n - 5)(n + 2)(n + 3)}$

Updated 2026-06-03

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