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The problem asks to perform the operation and simplify the expression:

\[
(4y - 1)^2 - (6y - 3)(6y + 3)
\]

### Step 1: Expand \((4y - 1)^2\)
This is a binomial square, so apply the formula \((a - b)^2 = a^2 - 2ab + b^2\):

\[
(4y - 1)^2 = (4y)^2 - 2(4y)(1) + 1^2 = 16y^2 - 8y + 1
\]

### Step 2: Expand \((6y - 3)(6y + 3)\)
This is a difference of squares, so use the identity \((a - b)(a + b) = a^2 - b^2\):

\[
(6y - 3)(6y + 3) = (6y)^2 - (3)^2 = 36y^2 - 9
\]

### Step 3: Subtract the two expanded expressions
Now substitute the expanded terms back into the original expression:

\[
(16y^2 - 8y + 1) - (36y^2 - 9)
\]

Distribute the negative sign across the second term:

\[
16y^2 - 8y + 1 - 36y^2 + 9
\]

### Step 4: Combine like terms
Now combine the terms:

\[
(16y^2 - 36y^2) + (-8y) + (1 + 9)
\]

This simplifies to:

\[
-20y^2 - 8y + 10
\]

### Final Answer:
\[
\boxed{-20y^2 - 8y + 10}
\]

Would you like any further details or clarifications?

Here are five related questions to expand your understanding:
1. How would you factor the expression \(-20y^2 - 8y + 10\)?
2. What is the general formula for expanding binomials?
3. How does the difference of squares formula apply in various algebraic contexts?
4. How can you verify the solution by substituting a value for \(y\)?
5. What are some common mistakes to avoid when simplifying algebraic expressions?

**Tip:** Always check for special factoring patterns like difference of squares or perfect square trinomials before expanding.

Learn how to simplify the expression (4y - 1)^2 - (6y - 3)(6y + 3) by applying algebraic identities such as the difference of squares and perfect square trinomial. This detailed solution is aimed at students in Grades 9-11.

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