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.