Divide the rational expressions

 

fraction numerator x squared plus 15 x plus 56 over denominator x squared plus 4 x minus 21 end fraction divided by fraction numerator x plus 8 over denominator x squared plus 5 x minus 24 end fraction

	a.	
left parenthesis x plus 8 right parenthesis left parenthesis x plus 7 right parenthesis

	b.	
fraction numerator left parenthesis x minus 3 right parenthesis left parenthesis x plus 7 right parenthesis over denominator x plus 8 end fraction

	c.	
x plus 8

	d.	
fraction numerator x plus 8 over denominator x minus 3 end fraction

We are asked to divide the following two rational expressions:

\[
\frac{x^2 + 15x + 56}{x^2 + 4x - 21} \div \frac{x + 8}{x^2 + 5x - 24}
\]

### Step 1: Rewrite the division as multiplication
To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction:

\[
\frac{x^2 + 15x + 56}{x^2 + 4x - 21} \times \frac{x^2 + 5x - 24}{x + 8}
\]

### Step 2: Factor each quadratic expression
Now, let's factor each of the polynomials involved.

- \(x^2 + 15x + 56\) factors as \((x + 7)(x + 8)\)
- \(x^2 + 4x - 21\) factors as \((x + 7)(x - 3)\)
- \(x^2 + 5x - 24\) factors as \((x + 8)(x - 3)\)

Thus, the expression becomes:

\[
\frac{(x + 7)(x + 8)}{(x + 7)(x - 3)} \times \frac{(x + 8)(x - 3)}{x + 8}
\]

### Step 3: Simplify
Now, cancel the common factors that appear in both the numerator and denominator:

- The factor \((x + 7)\) cancels.
- The factor \((x + 8)\) cancels.
- The factor \((x - 3)\) cancels.

After simplifying, we are left with:

\[
x + 8
\]

### Final Answer: 
The correct answer is:

\[
\boxed{c. \, x + 8}
\]

Would you like more details or further clarification?

Here are 5 related questions to consider:
1. How do you factor quadratic expressions?
2. What is the process for dividing rational expressions?
3. What happens when terms cancel out in a fraction?
4. How do you identify common factors in polynomials?
5. What are the rules for multiplying fractions?

**Tip:** Always check if expressions can be factored before simplifying rational expressions to make the process smoother!

Divide the rational expressions: (x^2 + 15x + 56) / (x^2 + 4x - 21) ÷ (x + 8) / (x^2 + 5x - 24)

Learn how to divide rational expressions by factoring quadratic expressions. This problem demonstrates key algebraic techniques, including multiplying by the reciprocal and simplifying common factors. Suitable for students in Grades 9-10.

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