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The Zero Product Property is a fundamental concept in algebra that helps us solve equations where the product of two or more terms equals zero. This property states that if the product of two factors is zero, then at least one of the factors must be zero.

For example, if we have the equation:
\( a \times b = 0 \),
then either \( a = 0 \), \( b = 0 \), or both must be zero.

In our given problem, \( 6y(4y+9)=0 \), we can use the Zero Product Property by setting each factor to zero independently.

- First factor: \( 6y = 0 \)
- Second factor: \( 4y + 9 = 0 \)

We solve each of these smaller equations to find the values of \( y \) that satisfy the original equation.

Factoring involves expressing an equation as a product of its simpler factors. This technique is instrumental in solving quadratic equations, especially when using the Zero Product Property.

In the problem \( 6y(4y+9)=0 \), the equation is already factored into two parts: \( 6y \) and \( 4y + 9 \).

Since these factors multiply to zero, we can individually set each to zero and solve for \( y \).

When solved:

• \( 6y = 0 \) gives \( y = 0 \).
• \( 4y + 9 = 0 \) simplifies to \( y = -\frac{9}{4} \).

Factoring is a powerful approach to decompose complex equations, making them easier to manage and solve systematically.

After finding potential solutions to an equation, it's vital to substitute them back into the original equation to verify their correctness.

Let's check the solutions of our given problem.

For \( y = 0 \):
Substitute \( y = 0 \) into the original equation:
\[ 6(0)(4(0)+9) = 0 \]
Simplifying, we get:
\[ 0 = 0 \]
This solution is correct.

For \( y = -\frac{9}{4} \):
Substitute \( y = -\frac{9}{4} \) into the original equation:
\[ 6\bigg(-\frac{9}{4}\bigg)\bigg(4\bigg(-\frac{9}{4}\bigg)+9\bigg) = 0 \]
Simplifying inside the parentheses:
\[ 6\bigg(-\frac{9}{4}\bigg)\times 0 = 0 \]
We get:
\[ 0 = 0 \]
This solution is also correct.

Always remember to check your solutions, as this confirms their validity and ensures that no mistakes were made during the solving process.