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.