We need to simplify the given expression:

\[
\frac{5y^2 - 28y - 12}{y + 5} \times \frac{y^2 + 10y + 25}{5y^2 + 27y + 10}
\]

### Step 1: Factor the Numerators and Denominators

1. **Factor \(5y^2 - 28y - 12\):**

   We need to factor this quadratic. The factors of \(5 \times -12 = -60\) that add up to \(-28\) are \(-30\) and \(2\).
   
   Split the middle term and factor by grouping:
   
   \[
   5y^2 - 30y + 2y - 12 = 5y(y - 6) + 2(y - 6) = (5y + 2)(y - 6)
   \]

2. **Factor \(y^2 + 10y + 25\):**

   This is a perfect square trinomial:
   
   \[
   y^2 + 10y + 25 = (y + 5)^2
   \]

3. **Factor \(5y^2 + 27y + 10\):**

   Again, we factor this quadratic. The factors of \(5 \times 10 = 50\) that add up to \(27\) are \(25\) and \(2\).
   
   Split the middle term and factor by grouping:
   
   \[
   5y^2 + 25y + 2y + 10 = 5y(y + 5) + 2(y + 5) = (5y + 2)(y + 5)
   \]

### Step 2: Write the Expression in Factored Form

Now, we rewrite the entire expression:

\[
\frac{(5y + 2)(y - 6)}{y + 5} \times \frac{(y + 5)^2}{(5y + 2)(y + 5)}
\]

### Step 3: Cancel Common Factors

We can cancel common factors from the numerator and denominator:

- The factor \(5y + 2\) cancels.
- One \(y + 5\) cancels.

This leaves us with:

\[
\frac{y - 6}{y + 5}
\]

### Final Answer:

The simplified expression is:

\[
\frac{y - 6}{y + 5}
\]

---

Would you like further details on any of these steps? Here are some related questions for practice:

1. How do you factor a quadratic equation of the form \(ax^2 + bx + c\)?
2. How do you recognize perfect square trinomials?
3. How does factoring by grouping work?
4. How do you identify common factors in rational expressions?
5. How do you handle division and multiplication of rational expressions?

**Tip:** Always check if a quadratic expression can be factored, as it simplifies the problem dramatically.

5y^2 − 28y − 12 / (y + 5) * (y^2 + 10y + 25) / (5y^2 + 27y + 10)

Learn how to simplify the rational expression 5y^2 - 28y - 12 / (y + 5) * (y^2 + 10y + 25) / (5y^2 + 27y + 10) with step-by-step factoring and common factor cancellation. This problem covers algebraic concepts such as factoring quadratics and simplifying rational expressions. Suitable for students in Grades 9-11.

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