Let's go through the expression step by step:

Given expression:

$(4y - 5)^2 - 3y(4y + 5) - y + 27$

Step 1: Expand the squared term

$(4y - 5)^2 = (4y - 5)(4y - 5)$ Using the distributive property (FOIL method): $(4y - 5)(4y - 5) = 16y^2 - 20y - 20y + 25 = 16y^2 - 40y + 25$

Step 2: Expand the second term

$- 3y(4y + 5) = -12y^2 - 15y$

Step 3: Substitute all the expanded terms back into the original expression:

$16y^2 - 40y + 25 - 12y^2 - 15y - y + 27$

Step 4: Combine like terms

• Combine $16y^2$ and $-12y^2$: $16y^2 - 12y^2 = 4y^2$
• Combine $-40y$, $-15y$, and $-y$: $-40y - 15y - y = -56y$
• Combine the constant terms $25$ and $27$: $25 + 27 = 52$

Thus, the simplified expression is: $4y^2 - 56y + 52$

Step 5: Factor the expression

Factor out the greatest common factor (GCF), which is 4: $4(y^2 - 14y + 13)$

Next, we factor the quadratic expression inside the parentheses: We are looking for two numbers that multiply to $13$ and add to $-14$. These numbers are $-13$ and $-1$.

Thus, we can factor the quadratic as: $4(y - 13)(y - 1)$

Final factored form:

$4(y - 13)(y - 1)$

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Would you like a more detailed breakdown of any of these steps? Here are a few related questions:

1. What is the process of factoring a quadratic expression in general?
2. How do you apply the distributive property (FOIL) when expanding binomials?
3. What happens if the greatest common factor is not immediately obvious?
4. Can you factor quadratics where the leading coefficient is not 1?
5. How can you check if your factored form is correct?

Tip: Always look for a common factor first, as it can simplify the factoring process for you!