The expression you provided is:

\[
\frac{y^2 - y - 30}{5y^2 - 20y - 60}
\]

Let's simplify this expression step by step.

### Step 1: Factor the numerator
The numerator is \( y^2 - y - 30 \). We need to factor this quadratic expression.

To factor \( y^2 - y - 30 \), we look for two numbers that multiply to \(-30\) and add to \(-1\) (the coefficient of \(y\)).

These numbers are \(-6\) and \(5\). So, we can factor the numerator as:

\[
y^2 - y - 30 = (y - 6)(y + 5)
\]

### Step 2: Factor the denominator
The denominator is \( 5y^2 - 20y - 60 \). We can start by factoring out the common factor of 5:

\[
5y^2 - 20y - 60 = 5(y^2 - 4y - 12)
\]

Now, let's factor the quadratic expression \( y^2 - 4y - 12 \).

We need to find two numbers that multiply to \(-12\) and add to \(-4\). These numbers are \(-6\) and \(2\). So, we can factor \( y^2 - 4y - 12 \) as:

\[
y^2 - 4y - 12 = (y - 6)(y + 2)
\]

Thus, the denominator becomes:

\[
5(y^2 - 4y - 12) = 5(y - 6)(y + 2)
\]

### Step 3: Simplify the expression
The simplified expression is:

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

We can cancel out the common factor \( (y - 6) \) in the numerator and denominator, assuming \( y \neq 6 \):

\[
\frac{y + 5}{5(y + 2)}
\]

This is the simplified form of the expression.

### Final Answer:
\[
\frac{y + 5}{5(y + 2)}
\]

Would you like more details or have any other questions?

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Here are 5 related questions you might find interesting:

1. How do you factor quadratic expressions in general?
2. What are the conditions under which terms can be canceled in a rational expression?
3. How do you solve equations involving rational expressions?
4. What happens if \( y = 6 \) in the original expression?
5. How would you simplify an expression with more complex factors in the numerator and denominator?

**Tip:** Always check for any restrictions on the variable when simplifying rational expressions, as canceling terms might not be valid for certain values.

Discover how to simplify the rational expression y^2 - y - 30 / 5y^2 - 20y - 60 through factoring and cancellation. This detailed step-by-step solution is ideal for students in Grades 9-10, covering algebraic concepts and the fundamental theorem of algebra.

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