Let's solve the equation step-by-step:

Given equation:

\[
3x^2 - 4(x + 2) = 0
\]

1. **Expand the equation**: Distribute the \(-4\) across the terms inside the parentheses:

\[
3x^2 - 4x - 8 = 0
\]

2. **Identify it as a quadratic equation**: This equation is in the form \(ax^2 + bx + c = 0\) where \(a = 3\), \(b = -4\), and \(c = -8\).

3. **Apply the quadratic formula**: The solutions to the quadratic equation \(ax^2 + bx + c = 0\) are given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Plugging in the values of \(a\), \(b\), and \(c\):

\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-8)}}{2(3)}
\]

4. **Simplify inside the square root**:

\[
x = \frac{4 \pm \sqrt{16 + 96}}{6}
\]

\[
x = \frac{4 \pm \sqrt{112}}{6}
\]

5. **Simplify the square root**:

\[
\sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7}
\]

So:

\[
x = \frac{4 \pm 4\sqrt{7}}{6}
\]

6. **Simplify the fractions**:

\[
x = \frac{4(1 \pm \sqrt{7})}{6} = \frac{2(1 \pm \sqrt{7})}{3}
\]

Therefore, the two solutions are:

\[
x_1 = \frac{2(1 + \sqrt{7})}{3}, \quad x_2 = \frac{2(1 - \sqrt{7})}{3}
\]

### Solutions:

\[
x_1 = \frac{2(1 + \sqrt{7})}{3}, \quad x_2 = \frac{2(1 - \sqrt{7})}{3}
\]

Would you like more details on any part of this solution? 

Here are 5 questions to expand your understanding:

1. How do you expand expressions with parentheses in algebra?
2. What are the steps to simplify a square root that involves multiplication?
3. How does the quadratic formula work, and when do you use it?
4. What does it mean when we say a quadratic equation has two solutions?
5. How do you rationalize the denominator of a fraction?

**Tip:** Always check your solutions by plugging them back into the original equation to verify they satisfy the equation.

Learn how to solve the equation 3x^2 - 4(x + 2) = 0 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the distributive property and the quadratic formula. Suitable for high school students in Grades 10-12.

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