Certainly! Here's a step-by-step explanation of all the formulas and techniques used to solve the quadratic equation by factoring.

### Given Equation
The given quadratic equation is:
\[
3q^2 + 4q + \frac{4}{3} = 0
\]

### Step 1: Eliminate the Fraction
To eliminate the fraction, multiply the entire equation by 3 (the denominator of the fraction). The formula applied here is:
\[
a(b + c + d) = ab + ac + ad
\]
In our case:
\[
3 \times (3q^2 + 4q + \frac{4}{3}) = 3 \times 0
\]

This simplifies to:
\[
9q^2 + 12q + 4 = 0
\]

### Step 2: Factoring the Quadratic Expression
The general form of a quadratic equation is:
\[
ax^2 + bx + c = 0
\]
Here, \( a = 9 \), \( b = 12 \), and \( c = 4 \).

To factor the quadratic expression \( 9q^2 + 12q + 4 \), we use the method of factoring by grouping. First, we find two numbers that multiply to \( a \times c \) (which is \( 9 \times 4 = 36 \)) and add to \( b \) (which is 12). 

The two numbers are 6 and 6 because:
\[
6 \times 6 = 36 \quad \text{and} \quad 6 + 6 = 12
\]

### Step 3: Split the Middle Term
We rewrite the middle term \( 12q \) as \( 6q + 6q \):
\[
9q^2 + 6q + 6q + 4 = 0
\]

### Step 4: Factor by Grouping
Group the terms in pairs:
\[
(9q^2 + 6q) + (6q + 4) = 0
\]

Factor out the greatest common factor (GCF) from each group:
\[
3q(3q + 2) + 2(3q + 2) = 0
\]
Here, the formula used is:
\[
ab + ac = a(b + c)
\]

### Step 5: Factor Out the Common Binomial
Since both groups have the common factor \( (3q + 2) \), we factor it out:
\[
(3q + 2)(3q + 2) = 0
\]

This simplifies to:
\[
(3q + 2)^2 = 0
\]

### Step 6: Solve for \( q \)
To solve for \( q \), use the Zero Product Property:
\[
\text{If } ab = 0, \text{ then } a = 0 \text{ or } b = 0.
\]
Apply this to our equation:
\[
3q + 2 = 0
\]

Now, solve for \( q \):
\[
3q = -2 \quad \Rightarrow \quad q = -\frac{2}{3}
\]

### Final Solution
The final solution is:
\[
q = -\frac{2}{3}
\]

Would you like any more explanations or have any further questions?

Here are some additional questions that might interest you:

1. How does the discriminant help in determining the nature of the roots of a quadratic equation?
2. What are the differences between factoring, completing the square, and using the quadratic formula?
3. How can you verify the solution to a quadratic equation after factoring?
4. What is the significance of the Zero Product Property in solving quadratic equations?
5. How do coefficients affect the factorability of a quadratic equation?

**Tip:** When factoring a quadratic equation, double-check your factors to ensure they multiply back to the original equation, confirming the correctness of your solution.

Explore a detailed step-by-step solution to solve the quadratic equation 3q^2 + 4q + 4/3 = 0 using factoring. Learn about factoring by grouping and the Zero Product Property. Suitable for high school students.

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