We are given the inequality:

\[
x^2 + 3x + 2 < 2
\]

### Step 1: Simplify the inequality
First, subtract 2 from both sides to simplify:

\[
x^2 + 3x + 2 - 2 < 0
\]

This reduces to:

\[
x^2 + 3x < 0
\]

### Step 2: Factor the left-hand side
We can factor the expression \( x^2 + 3x \) as follows:

\[
x(x + 3) < 0
\]

### Step 3: Solve for intervals
Now, we need to find where the product \( x(x + 3) \) is less than 0. This can be done by analyzing the signs of the factors.

1. The first factor, \( x \), is zero when \( x = 0 \).
2. The second factor, \( x + 3 \), is zero when \( x = -3 \).

These points, \( x = -3 \) and \( x = 0 \), divide the number line into three intervals:

1. \( x < -3 \)
2. \( -3 < x < 0 \)
3. \( x > 0 \)

### Step 4: Test the sign in each interval
We will test the sign of \( x(x + 3) \) in each interval:

1. For \( x < -3 \), both \( x \) and \( x + 3 \) are negative, so their product is positive.
2. For \( -3 < x < 0 \), \( x \) is negative and \( x + 3 \) is positive, so their product is negative.
3. For \( x > 0 \), both \( x \) and \( x + 3 \) are positive, so their product is positive.

### Step 5: Determine the solution
We are looking for where the product is less than 0. This occurs in the interval \( -3 < x < 0 \).

### Final answer:
The solution to the inequality is:

\[
-3 < x < 0
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How would the solution change if the inequality was \( x^2 + 3x + 2 \leq 2 \)?
2. What is the significance of factoring in solving inequalities?
3. How can you visualize the solution of quadratic inequalities on a number line?
4. How do inequalities behave when squared terms are involved?
5. Can you explain how the sign of a product helps determine solutions?

### Tip:
When solving quadratic inequalities, always look for points where the expression equals zero. These points help divide the number line into regions for sign analysis.

Learn how to solve the quadratic inequality x² + 3x + 2 < 2 with a detailed step-by-step solution. This problem covers important algebraic concepts, including factoring and sign analysis, suitable for students in Grades 9-11.

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