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Let's solve the quadratic equation \( 3q^2 + 4q + \frac{4}{3} = 0 \) by factoring.

### Step 1: Eliminate the fraction
To make factoring easier, let's eliminate the fraction by multiplying the entire equation by 3 (the denominator of the fraction):

\[
3 \times 3q^2 + 3 \times 4q + 3 \times \frac{4}{3} = 0
\]

This simplifies to:

\[
9q^2 + 12q + 4 = 0
\]

### Step 2: Factor the quadratic equation
Now, we need to factor the quadratic expression \( 9q^2 + 12q + 4 \).

We look for two numbers that multiply to the product of the coefficient of \( q^2 \) (which is 9) and the constant term (which is 4), and add to the coefficient of \( q \) (which is 12).

\[
9 \times 4 = 36
\]

We need two numbers that multiply to 36 and add to 12. These numbers are 6 and 6.

So, we can rewrite the middle term using 6 and 6:

\[
9q^2 + 6q + 6q + 4 = 0
\]

### Step 3: Group the terms and factor by grouping
Group the terms in pairs:

\[
(9q^2 + 6q) + (6q + 4) = 0
\]

Factor out the greatest common factor from each group:

\[
3q(3q + 2) + 2(3q + 2) = 0
\]

Now, factor out the common binomial factor \( 3q + 2 \):

\[
(3q + 2)(3q + 2) = 0
\]

This simplifies to:

\[
(3q + 2)^2 = 0
\]

### Step 4: Solve for \( q \)
To solve for \( q \), set the factor equal to zero:

\[
3q + 2 = 0
\]

Solve for \( q \):

\[
3q = -2
\]

\[
q = -\frac{2}{3}
\]

### Final Answer:
The solution is \( q = -\frac{2}{3} \).

Would you like more details or have any questions about the solution? 

Here are some related questions:

1. How can you check the solution to a factored quadratic equation?
2. What methods can be used to solve quadratic equations besides factoring?
3. How does completing the square differ from factoring when solving a quadratic equation?
4. Why is it useful to multiply out to eliminate fractions before factoring a quadratic equation?
5. What are the characteristics of quadratic equations that make them factorable?

**Tip:** When factoring quadratic equations, it's often helpful to first simplify the equation by eliminating fractions or combining like terms.

Learn how to solve the quadratic equation 3q^2 + 4q + 4/3 by factoring with a detailed step-by-step solution. This problem covers key algebraic concepts, including quadratic equations and factoring methods. Suitable for high school students.

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