perform the indicated operation and simplify the result. leave your answer in factored form.\n$\frac{x}{x +…

Explanation:

Step1: Find common denominator

The common denominator of $x + 9$ and $x-9$ is $(x + 9)(x - 9)$. So we rewrite the fractions: $\frac{x}{x + 9}\cdot\frac{x - 9}{x - 9}+\frac{8x-5}{x - 9}\cdot\frac{x + 9}{x + 9}=\frac{x(x - 9)}{(x + 9)(x - 9)}+\frac{(8x - 5)(x + 9)}{(x + 9)(x - 9)}$.

Step2: Expand numerators

Expand $x(x - 9)=x^{2}-9x$ and $(8x - 5)(x + 9)=8x^{2}+72x-5x - 45=8x^{2}+67x - 45$. Then the expression becomes $\frac{x^{2}-9x+(8x^{2}+67x - 45)}{(x + 9)(x - 9)}$.

Step3: Combine like - terms in numerator

Combine like - terms: $x^{2}-9x+8x^{2}+67x - 45=(x^{2}+8x^{2})+(-9x + 67x)-45=9x^{2}+58x - 45$.

Step4: Factor the numerator

Factor $9x^{2}+58x - 45$. We need to find two numbers $a$ and $b$ such that $a\times b=9\times(-45)=-405$ and $a + b=58$. The numbers are $67$ and $- 9$. Then $9x^{2}+58x - 45=9x^{2}+67x-9x - 45=(9x^{2}+67x)-(9x + 45)=x(9x + 67)-5(9x + 9)=(9x - 5)(x + 9)$. The final expression is $\frac{(9x - 5)(x + 9)}{(x + 9)(x - 9)}$.

Step5: Simplify the fraction

Cancel out the common factor $(x + 9)$ (assuming $x\neq - 9$). The simplified result is $\frac{9x - 5}{x - 9}$.

Answer:

$\frac{9x - 5}{x - 9}$