To solve the quadratic equation \( 2y^2 + 7y + 5 = 0 \) by factoring, follow these steps:

### 1. Identify the quadratic expression:
The quadratic expression is in the form \( ax^2 + bx + c \), where:
- \( a = 2 \)
- \( b = 7 \)
- \( c = 5 \)

### 2. Multiply \( a \) and \( c \):
Multiply the coefficient of \( y^2 \) (which is \( a = 2 \)) by the constant term \( c = 5 \):
\[
2 \times 5 = 10
\]
We need to find two numbers that multiply to \( 10 \) and add to \( 7 \) (the coefficient of \( y \)).

### 3. Find the pair of factors:
The pair of factors that multiply to \( 10 \) and add to \( 7 \) are \( 2 \) and \( 5 \):
\[
2 \times 5 = 10
\quad \text{and} \quad
2 + 5 = 7
\]

### 4. Rewrite the middle term using the factors:
Rewrite the middle term \( 7y \) as the sum of two terms using the factors \( 2y \) and \( 5y \):
\[
2y^2 + 2y + 5y + 5 = 0
\]

### 5. Factor by grouping:
Now, group the terms in pairs and factor out the common factors:
\[
2y(y + 1) + 5(y + 1) = 0
\]
Factor out the common binomial factor \( (y + 1) \):
\[
(2y + 5)(y + 1) = 0
\]

### 6. Solve for \( y \):
Set each factor equal to zero and solve for \( y \):
\[
2y + 5 = 0 \quad \text{or} \quad y + 1 = 0
\]
For \( 2y + 5 = 0 \):
\[
2y = -5
\quad \Rightarrow \quad
y = -\frac{5}{2}
\]
For \( y + 1 = 0 \):
\[
y = -1
\]

### Final Answer:
The solutions are \( y = -\frac{5}{2} \) and \( y = -1 \).

Would you like any further details or explanations on this problem?

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Here are five related questions:
1. How would you solve \( 3y^2 + 11y + 6 = 0 \) by factoring?
2. What if the quadratic equation was \( 4y^2 + 4y + 1 = 0 \)? How would you factor and solve it?
3. Can you solve the quadratic equation by completing the square instead of factoring?
4. How do you determine the number of real solutions to a quadratic equation using the discriminant?
5. What are the roots of the quadratic equation \( y^2 - 6y + 9 = 0 \)?

**Tip:** When factoring quadratics, always check if the leading coefficient is 1. If not, make sure to factor it in before proceeding with grouping.

Discover how to solve the quadratic equation 2y^2 + 7y + 5 = 0 by factoring with a comprehensive step-by-step guide. This lesson covers essential algebraic concepts, including factoring techniques and the application of the Zero Product Property. Perfect for students in Grades 8-10.

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