To simplify the expression \( \frac{4x^2 - 24x + 36}{5x^2 - 6x - 27} \), let's factor both the numerator and the denominator.

### Step 1: Simplify the numerator
The numerator is \( 4x^2 - 24x + 36 \). We can factor out the common factor of 4:
\[
4x^2 - 24x + 36 = 4(x^2 - 6x + 9)
\]
Now, let's factor the quadratic \( x^2 - 6x + 9 \).

Notice that \( x^2 - 6x + 9 \) is a perfect square trinomial:
\[
x^2 - 6x + 9 = (x - 3)^2
\]
Thus, the numerator becomes:
\[
4(x - 3)^2
\]

### Step 2: Simplify the denominator
The denominator is \( 5x^2 - 6x - 27 \). We need to factor this quadratic expression. 

To factor \( 5x^2 - 6x - 27 \), we look for two numbers that multiply to \( 5 \times -27 = -135 \) and add to -6. The numbers 9 and -15 work:
\[
5x^2 - 6x - 27 = 5x^2 + 9x - 15x - 27
\]
Now, factor by grouping:
\[
= (5x^2 + 9x) - (15x + 27)
\]
\[
= x(5x + 9) - 3(5x + 9)
\]
\[
= (x - 3)(5x + 9)
\]
So, the denominator becomes:
\[
(5x + 9)(x - 3)
\]

### Step 3: Simplify the expression
Now, substitute the factored forms back into the original expression:
\[
\frac{4(x - 3)^2}{(5x + 9)(x - 3)}
\]
We can cancel the common factor \( x - 3 \) from the numerator and denominator:
\[
\frac{4(x - 3)}{5x + 9}
\]

### Final Answer:
The simplified expression is:
\[
\frac{4(x - 3)}{5x + 9}
\]

Would you like any further details or have any questions about this solution?

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Here are 5 questions related to the topic:
1. How would the expression change if the quadratic in the numerator was not a perfect square?
2. What if the denominator had a different quadratic expression, like \( 5x^2 - 7x - 27 \)?
3. How can you check if the factorization of the denominator is correct?
4. What if the original expression had a negative sign in front of it?
5. How would you solve for \( x \) if the expression were set equal to a number?

**Tip:** Always look for common factors in quadratic expressions before starting to factor. This simplifies the problem significantly.

Learn how to simplify the expression 4x^2 - 24x + 36 / 5x^2 - 6x - 27 with a detailed step-by-step solution. This problem involves factoring quadratic expressions in the numerator and denominator. Suitable for high school students in Grades 9-12.

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