Answer :
To find the sum of the rational expressions [tex]\(\frac{x}{x-2} + \frac{7x}{x+5}\)[/tex], let's break the problem down into manageable steps.
### Step-by-Step Solution
Step 1: Identify the rational expressions.
We have:
[tex]\[ \frac{x}{x-2} \quad \text{and} \quad \frac{7x}{x+5} \][/tex]
Step 2: Find a common denominator.
The common denominator of [tex]\(x-2\)[/tex] and [tex]\(x+5\)[/tex] is:
[tex]\[ (x-2)(x+5) \][/tex]
Step 3: Rewrite each fraction with the common denominator.
Rewrite [tex]\(\frac{x}{x-2}\)[/tex]:
[tex]\[ \frac{x}{x-2} = \frac{x(x+5)}{(x-2)(x+5)} = \frac{x^2 + 5x}{(x-2)(x+5)} \][/tex]
Rewrite [tex]\(\frac{7x}{x+5}\)[/tex]:
[tex]\[ \frac{7x}{x+5} = \frac{7x(x-2)}{(x-2)(x+5)} = \frac{7x^2 - 14x}{(x-2)(x+5)} \][/tex]
Step 4: Add the rewritten fractions.
Now add the fractions:
[tex]\[ \frac{x^2 + 5x}{(x-2)(x+5)} + \frac{7x^2 - 14x}{(x-2)(x+5)} = \frac{(x^2 + 5x) + (7x^2 - 14x)}{(x-2)(x+5)} \][/tex]
Combine the numerators:
[tex]\[ \frac{x^2 + 5x + 7x^2 - 14x}{(x-2)(x+5)} = \frac{8x^2 - 9x}{(x-2)(x+5)} \][/tex]
Step 5: Simplify the resulting expression.
The expression we end up with is:
[tex]\[ \frac{8x^2 - 9x}{(x-2)(x+5)} \][/tex]
Step 6: Match the result with the given choices.
Among the given options, the correct answer is:
D. [tex]\(\frac{8x^2-9x}{x^2+3x-10}\)[/tex]
Note that [tex]\(x^2 + 3x - 10\)[/tex] is the expanded form of [tex]\((x-2)(x+5)\)[/tex].
### Final Answer:
The sum of the given rational expressions is:
[tex]\[ \boxed{\frac{8x^2 - 9x}{x^2 + 3x - 10}} \][/tex]
### Step-by-Step Solution
Step 1: Identify the rational expressions.
We have:
[tex]\[ \frac{x}{x-2} \quad \text{and} \quad \frac{7x}{x+5} \][/tex]
Step 2: Find a common denominator.
The common denominator of [tex]\(x-2\)[/tex] and [tex]\(x+5\)[/tex] is:
[tex]\[ (x-2)(x+5) \][/tex]
Step 3: Rewrite each fraction with the common denominator.
Rewrite [tex]\(\frac{x}{x-2}\)[/tex]:
[tex]\[ \frac{x}{x-2} = \frac{x(x+5)}{(x-2)(x+5)} = \frac{x^2 + 5x}{(x-2)(x+5)} \][/tex]
Rewrite [tex]\(\frac{7x}{x+5}\)[/tex]:
[tex]\[ \frac{7x}{x+5} = \frac{7x(x-2)}{(x-2)(x+5)} = \frac{7x^2 - 14x}{(x-2)(x+5)} \][/tex]
Step 4: Add the rewritten fractions.
Now add the fractions:
[tex]\[ \frac{x^2 + 5x}{(x-2)(x+5)} + \frac{7x^2 - 14x}{(x-2)(x+5)} = \frac{(x^2 + 5x) + (7x^2 - 14x)}{(x-2)(x+5)} \][/tex]
Combine the numerators:
[tex]\[ \frac{x^2 + 5x + 7x^2 - 14x}{(x-2)(x+5)} = \frac{8x^2 - 9x}{(x-2)(x+5)} \][/tex]
Step 5: Simplify the resulting expression.
The expression we end up with is:
[tex]\[ \frac{8x^2 - 9x}{(x-2)(x+5)} \][/tex]
Step 6: Match the result with the given choices.
Among the given options, the correct answer is:
D. [tex]\(\frac{8x^2-9x}{x^2+3x-10}\)[/tex]
Note that [tex]\(x^2 + 3x - 10\)[/tex] is the expanded form of [tex]\((x-2)(x+5)\)[/tex].
### Final Answer:
The sum of the given rational expressions is:
[tex]\[ \boxed{\frac{8x^2 - 9x}{x^2 + 3x - 10}} \][/tex]
