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StartFraction x squared minus 7 x Over x squared plus 5 x EndFraction plus StartFraction x squared plus 4 x Over x squared plus 5 x EndFractionx2−7xx2+5x+x2+4xx2+5x

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Part 1

Select the correct choice below and fill in any answer boxes within your choice.

​(Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.)

A.

StartFraction x squared minus 7 x Over x squared plus 5 x EndFraction plus StartFraction x squared plus 4 x Over x squared plus 5 x EndFraction equals nothingx2−7xx2+5x+x2+4xx2+5x=enter your response here​, 

x not equals nothingx≠enter your response here 

B.

StartFraction x squared minus 7 x Over x squared plus 5 x EndFraction plus StartFraction x squared plus 4 x Over x squared plus 5 x EndFraction equalsx2−7xx2+5x+x2+4xx2+5x=enter your response here​, 

no numbers must be excluded.

Let's add the given rational expressions:

\[
\frac{x^2 - 7x}{x^2 + 5x} + \frac{x^2 + 4x}{x^2 + 5x}
\]

### Step 1: Identify the common denominator

Both fractions have the same denominator \(x^2 + 5x\). This allows us to directly add the numerators while keeping the common denominator.

### Step 2: Add the numerators

Combine the numerators of the fractions:

\[
(x^2 - 7x) + (x^2 + 4x) = x^2 - 7x + x^2 + 4x
\]

Simplify the expression:

\[
x^2 + x^2 - 7x + 4x = 2x^2 - 3x
\]

### Step 3: Write the resulting expression

The resulting expression is:

\[
\frac{2x^2 - 3x}{x^2 + 5x}
\]

### Step 4: Simplify the expression (if possible)

Now, let's factor both the numerator and denominator:

1. Factor the numerator \(2x^2 - 3x\):
   \[
   2x^2 - 3x = x(2x - 3)
   \]

2. Factor the denominator \(x^2 + 5x\):
   \[
   x^2 + 5x = x(x + 5)
   \]

So, the expression simplifies to:

\[
\frac{x(2x - 3)}{x(x + 5)}
\]

We can cancel the common factor \(x\) in the numerator and denominator, provided \(x \neq 0\):

\[
\frac{2x - 3}{x + 5}
\]

### Step 5: Determine when the expression is undefined

The expression is undefined when the original denominator \(x^2 + 5x\) is zero:

\[
x(x + 5) = 0
\]

This gives \(x = 0\) or \(x = -5\).

### Final Answer:

The simplified expression is:

\[
\frac{2x - 3}{x + 5}, \quad x \neq 0, -5
\]

The correct choice is:

**A**  
\[
\frac{2x - 3}{x + 5}, \quad x \neq 0, -5
\]

Would you like more details or have any questions?

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Here are some related questions you might consider:

1. How do you add and subtract rational expressions?
2. What is the process for factoring polynomials?
3. How do you determine when a rational expression is undefined?
4. What are the steps to simplify complex fractions?
5. How do you handle rational expressions with different denominators?

**Tip**: Always factor the numerator and denominator to identify common factors that can be canceled to simplify the expression.

Learn how to add rational expressions (x^2 - 7x)/(x^2 + 5x) and (x^2 + 4x)/(x^2 + 5x) with a step-by-step solution. This problem covers algebraic concepts such as rational expressions and fraction addition. Suitable for high school students.

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