1. **State the problem:** Solve the rational equation $$1 + \frac{2}{x+1} = \frac{3x+7}{x^2 + 10x + 9}$$ for all values of $x$. 2. **Identify the denominators and restrictions:** The denominators are $x+1$ and $x^2 + 10x + 9$. Factor the quadratic denominator: $$x^2 + 10x + 9 = (x+1)(x+9)$$ Restrictions: $x \neq -1$ and $x \neq -9$ because these values make denominators zero. 3. **Rewrite the equation with factored denominators:** $$1 + \frac{2}{x+1} = \frac{3x+7}{(x+1)(x+9)}$$ 4. **Multiply both sides by the common denominator $(x+1)(x+9)$ to clear fractions:** $$ (x+1)(x+9) \cdot 1 + (x+1)(x+9) \cdot \frac{2}{x+1} = (x+1)(x+9) \cdot \frac{3x+7}{(x+1)(x+9)} $$ Simplify each term: - First term: $(x+1)(x+9)$ - Second term: $(x+9) \cdot 2 = 2(x+9)$ - Third term: $3x+7$ So the equation becomes: $$ (x+1)(x+9) + 2(x+9) = 3x + 7 $$ 5. **Expand and simplify:** - Expand $(x+1)(x+9) = x^2 + 10x + 9$ - Expand $2(x+9) = 2x + 18$ So: $$ x^2 + 10x + 9 + 2x + 18 = 3x + 7 $$ Combine like terms on the left: $$ x^2 + 12x + 27 = 3x + 7 $$ 6. **Bring all terms to one side to form a quadratic equation:** $$ x^2 + 12x + 27 - 3x - 7 = 0 $$ $$ x^2 + 9x + 20 = 0 $$ 7. **Factor the quadratic:** $$ x^2 + 9x + 20 = (x + 4)(x + 5) = 0 $$ 8. **Solve for $x$:** $$ x + 4 = 0 \Rightarrow x = -4 $$ $$ x + 5 = 0 \Rightarrow x = -5 $$ 9. **Check restrictions:** Recall $x \neq -1$ and $x \neq -9$. Both $-4$ and $-5$ are allowed. **Final answer:** $$ \boxed{x = -4, -5} $$