To solve the inequality \( x^2 + 2x^2 + x + 2 > 0 \), let's first simplify the expression:

### Step 1: Combine like terms
\[
x^2 + 2x^2 + x + 2 = 3x^2 + x + 2
\]

So, the inequality becomes:
\[
3x^2 + x + 2 > 0
\]

### Step 2: Analyze the quadratic expression
The inequality \( 3x^2 + x + 2 > 0 \) is a quadratic inequality. The expression is in the form \( ax^2 + bx + c \), where \( a = 3 \), \( b = 1 \), and \( c = 2 \).

### Step 3: Determine the discriminant
The discriminant \( \Delta \) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[
\Delta = b^2 - 4ac
\]
Substituting the values of \( a \), \( b \), and \( c \):
\[
\Delta = 1^2 - 4(3)(2) = 1 - 24 = -23
\]

### Step 4: Analyze the discriminant
Since the discriminant \( \Delta \) is negative (\( -23 \)), the quadratic equation \( 3x^2 + x + 2 = 0 \) has no real roots. This implies that the parabola described by \( y = 3x^2 + x + 2 \) does not cross the x-axis and is always either above or below the x-axis.

### Step 5: Determine the sign of the parabola
Because the coefficient of \( x^2 \) (which is 3) is positive, the parabola opens upwards. Therefore, \( 3x^2 + x + 2 \) is always positive for all real values of \( x \).

### Conclusion
The inequality \( 3x^2 + x + 2 > 0 \) is true for all real numbers \( x \).

Would you like further details or have any questions?

### Related Questions:
1. How do you solve a quadratic inequality with real roots?
2. What is the significance of the discriminant in quadratic equations?
3. How do you determine the intervals where a quadratic function is positive or negative?
4. What is the role of the leading coefficient in determining the direction of a parabola?
5. Can you explain how to factorize quadratic expressions?

**Tip:** For any quadratic expression \( ax^2 + bx + c \), the sign of \( a \) determines whether the parabola opens upwards or downwards.

Learn how to solve the inequality 3x^2 + x + 2 > 0 with a detailed step-by-step solution. This problem covers quadratic equations, inequalities, and the discriminant formula. Suitable for students in Grades 10-12.

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