Question content area top
Part 1
Subtract.
StartFraction x plus 3 Over x squared minus 4 EndFraction minus StartFraction x plus 5 Over x minus 2 EndFraction
Question content area bottom
Part 1
Select the correct choice below and fill in any answer​ box(es) 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 plus 3 Over x squared minus 4 EndFraction minus StartFraction x plus 5 Over x minus 2 EndFraction
equals
  
enter your response here​, xnot equals
  
enter your response here
B.
StartFraction x plus 3 Over x squared minus 4 EndFraction minus StartFraction x plus 5 Over x minus 2 EndFraction
equals
  
enter your response here​, no numbers must be excluded.

Let's start by simplifying the given expression:

\[
\frac{x + 3}{x^2 - 4} - \frac{x + 5}{x - 2}
\]

### Step 1: Factor the denominators
The first denominator, \(x^2 - 4\), can be factored using the difference of squares:

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

So the expression becomes:

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

### Step 2: Find the common denominator
The common denominator for both fractions is \((x - 2)(x + 2)\). 

### Step 3: Rewrite the second fraction
To get the common denominator for the second fraction, multiply both the numerator and the denominator of the second fraction by \((x + 2)\):

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

### Step 4: Subtract the fractions
Now, subtract the two fractions:

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

### Step 5: Expand and simplify the numerator
Expand the numerator:

\[
(x + 5)(x + 2) = x^2 + 2x + 5x + 10 = x^2 + 7x + 10
\]

So the numerator becomes:

\[
(x + 3) - (x^2 + 7x + 10)
\]

Simplify the numerator:

\[
-x^2 - 6x - 7
\]

### Step 6: Write the final simplified expression

\[
\frac{-x^2 - 6x - 7}{(x - 2)(x + 2)}
\]

### Step 7: Determine excluded values
The excluded values occur when the denominator is equal to zero. For \((x - 2)(x + 2) = 0\):

\[
x - 2 = 0 \quad \text{or} \quad x + 2 = 0
\]
\[
x = 2 \quad \text{or} \quad x = -2
\]

### Final Answer:
The expression simplifies to:

\[
\frac{-x^2 - 6x - 7}{(x - 2)(x + 2)}, \quad x \neq 2, -2
\]

So the correct choice is:

**Choice A**: 
\[
\frac{-x^2 - 6x - 7}{(x - 2)(x + 2)}, \quad x \neq 2, -2
\]

Would you like further details or explanations? Here are some related questions to consider:

1. How do you factor quadratic expressions like \(x^2 - 4\)?
2. What are the steps for finding a common denominator in rational expressions?
3. Can you explain why we exclude certain values from the domain of the expression?
4. What is the difference between simplifying and solving an expression?
5. How do you expand a binomial like \((x + 5)(x + 2)\)?

**Tip:** Always check the denominator for values that could make the expression undefined by setting it equal to zero and solving for the variable.

Learn how to simplify the rational expression (x + 3)/(x^2 - 4) - (x + 5)/(x - 2) step-by-step. This problem covers algebraic concepts such as rational expressions, factoring, and excluded values. Suitable for high school students.

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