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To solve the problem, we need to find the angle that the line \( 2x + 3y + 4 = 0 \) makes with the positive x-axis.

### Steps:

1. **Find the slope of the line:**
   The equation of the line is given in general form \( Ax + By + C = 0 \). To convert it to slope-intercept form \( y = mx + b \), solve for \( y \):

   \[
   2x + 3y + 4 = 0
   \]
   \[
   3y = -2x - 4
   \]
   \[
   y = -\frac{2}{3}x - \frac{4}{3}
   \]

   The slope \( m \) is \( -\frac{2}{3} \).

2. **Use the slope to find the angle:**
   The angle \( \theta \) that the line makes with the positive x-axis can be found using the formula:
   \[
   \tan(\theta) = |m|
   \]
   \[
   \tan(\theta) = \left| -\frac{2}{3} \right| = \frac{2}{3}
   \]

   Taking the arctangent of both sides:
   \[
   \theta = \tan^{-1}\left(\frac{2}{3}\right)
   \]

3. **Calculate the angle:**

   Let me compute this.

The angle \( \theta \) that the line makes with the positive x-axis is approximately \( 33.69^\circ \), which is closest to option A: \( 34^\circ \).

Would you like more details on this solution or have any other questions?

Here are five related questions to explore:
1. How do you derive the slope-intercept form from the general form of a line?
2. What is the relationship between slope and angle for a line?
3. How can you calculate the angle if the slope is negative or positive?
4. Why is the arctangent function used to find the angle from the slope?
5. How would the answer change if the slope were steeper?

**Tip:** For any line, the angle with the x-axis can be found directly from its slope using the inverse tangent function.

The equation 2x + 3y + 4 = 0 makes an angle with the positive x-axis. Which of the following is closest to the size of the angle? A. 34° B. 56° C. 124° D. 146°

Learn how to calculate the angle that the line 2x + 3y + 4 = 0 makes with the positive x-axis. This problem covers key algebraic and trigonometric concepts, including finding the slope of the line and using the arctangent function to determine the angle.

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