To factor the quadratic expression \( 5y^2 + 8y + 3 \), we will use the method of "splitting the middle term" or the "AC method." Let's go step by step:

### Step 1: Multiply the coefficient of \( y^2 \) (which is 5) by the constant term (which is 3):
\[
5 \times 3 = 15
\]

### Step 2: Find two numbers that multiply to give 15 and add up to 8 (the coefficient of \( y \)):
The two numbers are 3 and 5 because:
\[
3 \times 5 = 15 \quad \text{and} \quad 3 + 5 = 8
\]

### Step 3: Rewrite the middle term (8y) using the two numbers found:
\[
5y^2 + 3y + 5y + 3
\]

### Step 4: Group the terms in pairs:
\[
(5y^2 + 3y) + (5y + 3)
\]

### Step 5: Factor each group:
\[
y(5y + 3) + 1(5y + 3)
\]

### Step 6: Factor out the common binomial \( (5y + 3) \):
\[
(5y + 3)(y + 1)
\]

So, the factorized form of \( 5y^2 + 8y + 3 \) is:
\[
(5y + 3)(y + 1)
\]

Would you like further details or clarification?

### Related Questions:
1. How would you factor a quadratic where the leading coefficient is 1?
2. Can you factor the expression \( 6y^2 + 7y + 2 \)?
3. What is the difference between factoring and solving a quadratic equation?
4. How do you use the quadratic formula to solve \( 5y^2 + 8y + 3 = 0 \)?
5. Can you apply factoring to higher degree polynomials?

### Tip:
Always check your factorization by expanding the factored form to ensure it matches the original expression.

Step-by-step guide to factor the quadratic expression 5y^2 + 8y + 3 using the method of splitting the middle term. Learn how to break down the equation into simpler terms and factor it into (5y + 3)(y + 1). Suitable for students in Grades 8-10.

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