https://file.aiquickdraw.com/mathbot/file/1726198114448izz2m2ld.jpg

The image contains two expressions that need to be simplified. Let's go through both steps one by one:

### Expression (a):
\[
\frac{15x + 5}{75x^2 + 25x}
\]

**Step 1: Factor the numerator and denominator.**
- The numerator \( 15x + 5 \) can be factored as \( 5(3x + 1) \).
- The denominator \( 75x^2 + 25x \) can be factored as \( 25x(3x + 1) \).

Now the expression becomes:
\[
\frac{5(3x + 1)}{25x(3x + 1)}
\]

**Step 2: Cancel common terms.**
Since \( (3x + 1) \) appears in both the numerator and denominator, we can cancel them:
\[
\frac{5}{25x}
\]

Simplify further:
\[
\frac{1}{5x}
\]

So, the simplified form of (a) is:
\[
\frac{1}{5x}
\]

---

### Expression (b):
\[
\frac{27x^2 + 9x}{24x^2 + 8x}
\]

**Step 1: Factor the numerator and denominator.**
- The numerator \( 27x^2 + 9x \) can be factored as \( 9x(3x + 1) \).
- The denominator \( 24x^2 + 8x \) can be factored as \( 8x(3x + 1) \).

Now the expression becomes:
\[
\frac{9x(3x + 1)}{8x(3x + 1)}
\]

**Step 2: Cancel common terms.**
Since \( (3x + 1) \) appears in both the numerator and denominator, we can cancel them:
\[
\frac{9x}{8x}
\]

**Step 3: Simplify the remaining expression.**
Cancel \( x \) from both the numerator and denominator:
\[
\frac{9}{8}
\]

So, the simplified form of (b) is:
\[
\frac{9}{8}
\]

---

### Final Answer:
- (a) \( \frac{1}{5x} \)
- (b) \( \frac{9}{8} \)

Let me know if you need further details or explanations.

Here are some follow-up questions:
1. What would happen if you had more complex terms in the numerator and denominator?
2. How do you factor quadratic expressions that have no common factors?
3. What is the importance of recognizing common factors in simplifying rational expressions?
4. Can you apply these steps to polynomials of higher degree?
5. How can you verify your simplifications are correct?

**Tip:** Always look for common factors in both the numerator and denominator to simplify expressions quickly!

Simplify the given expressions:
(a) (15x + 5) / (75x^2 + 25x)
(b) (27x^2 + 9x) / (24x^2 + 8x)

Learn how to simplify rational expressions by factoring and cancelling common terms. This problem involves simplifying two rational expressions step by step, including factoring the numerator and denominator. Suitable for students in Grades 9-11.

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