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Part 1
Factor the​ trinomial, or state that the trinomial is prime.
4 y squared plus 24 y plus 11
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Part 1
Select the correct choice below and fill in any answer boxes within your choice.
A.4 y squared plus 24 y plus 11 equals enter your response here
4 y squared plus 24 y plus 11equals
  
enter your response here

We need to **factor the trinomial** \( 4y^2 + 24y + 11 \). Let's walk through the steps carefully to determine if it can be factored.

### Step 1: Identify the coefficients
The trinomial is in the form \( ax^2 + bx + c \), where:
- \( a = 4 \)
- \( b = 24 \)
- \( c = 11 \)

### Step 2: Use the AC Method (if necessary)
To factor the trinomial, we use the **AC Method**, where:
1. Multiply \( a \times c = 4 \times 11 = 44 \).
2. Now, we need two numbers that multiply to 44 and add up to \( b = 24 \). These numbers are **2** and **22**.

### Step 3: Rewrite the middle term using these numbers
Rewrite \( 24y \) as \( 2y + 22y \):
\[
4y^2 + 2y + 22y + 11
\]

### Step 4: Factor by grouping
Group the terms in pairs:
\[
(4y^2 + 2y) + (22y + 11)
\]

Now, factor out the greatest common factor (GCF) from each group:
\[
2y(2y + 1) + 11(2y + 1)
\]

### Step 5: Factor out the common binomial
Both terms have a common factor of \( (2y + 1) \):
\[
(2y + 1)(2y + 11)
\]

### Final Answer
Thus, the factorization of \( 4y^2 + 24y + 11 \) is:
\[
(2y + 1)(2y + 11)
\]

---

**Answer choice:**
\[
4y^2 + 24y + 11 = (2y + 1)(2y + 11)
\]

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Do you need further clarification, or have any other questions?  

Here are 5 related questions:
1. How do you know if a trinomial is **prime**?
2. What is the **AC method**, and when should it be used?
3. How do you factor trinomials when the leading coefficient \(a\) is **1**?
4. What other methods can be used to factor **quadratic trinomials**?
5. Can you always use **grouping** to factor polynomials?

**Tip:** If a trinomial is hard to factor, checking its discriminant \(b^2 - 4ac\) can tell you if it has real factors.

Learn how to factor the quadratic trinomial 4y^2 + 24y + 11 using the AC Method and factoring by grouping. This detailed solution explains step-by-step how to break down the middle term and group terms for factoring. Perfect for students in Grades 9-10.

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