We are asked to solve the inequality:

$3x^2 + 10x + 5 \geq 2$

Step 1: Move all terms to one side

We start by subtracting 2 from both sides of the inequality:

$3x^2 + 10x + 5 - 2 \geq 0$

This simplifies to:

$3x^2 + 10x + 3 \geq 0$

Step 2: Solve the corresponding equation

To understand the nature of the inequality, let's first solve the related quadratic equation:

$3x^2 + 10x + 3 = 0$

We will use the quadratic formula:

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

where $a = 3$, $b = 10$, and $c = 3$. Substituting these values into the quadratic formula:

$x = \frac{-10 \pm \sqrt{10^2 - 4(3)(3)}}{2(3)}$

$x = \frac{-10 \pm \sqrt{100 - 36}}{6}$

$x = \frac{-10 \pm \sqrt{64}}{6}$

$x = \frac{-10 \pm 8}{6}$

This gives two solutions:

$x = \frac{-10 + 8}{6} = \frac{-2}{6} = -\frac{1}{3}$

and

$x = \frac{-10 - 8}{6} = \frac{-18}{6} = -3$

Thus, the solutions to the equation $3x^2 + 10x + 3 = 0$ are $x = -3$ and $x = -\frac{1}{3}$.

Step 3: Determine the intervals for the inequality

The solutions $x = -3$ and $x = -\frac{1}{3}$ divide the real number line into three intervals:

1. $(-\infty, -3)$
2. $(-3, -\frac{1}{3})$
3. $(-\frac{1}{3}, \infty)$

We now test the inequality $3x^2 + 10x + 3 \geq 0$ in each of these intervals.

Step 4: Test the intervals

• For $x < -3$ (e.g., $x = -4$): $3(-4)^2 + 10(-4) + 3 = 3(16) - 40 + 3 = 48 - 40 + 3 = 11 \quad (\text{positive})$

• For $-3 < x < -\frac{1}{3}$ (e.g., $x = -1$): $3(-1)^2 + 10(-1) + 3 = 3(1) - 10 + 3 = 3 - 10 + 3 = -4 \quad (\text{negative})$

• For $x > -\frac{1}{3}$ (e.g., $x = 0$): $3(0)^2 + 10(0) + 3 = 0 + 0 + 3 = 3 \quad (\text{positive})$

Step 5: Conclusion

From the tests above, we see that the inequality $3x^2 + 10x + 3 \geq 0$ is satisfied when $x \leq -3$ or $x \geq -\frac{1}{3}$.

Thus, the solution to the inequality is:

$x \in (-\infty, -3] \cup \left[-\frac{1}{3}, \infty\right)$

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Related questions:

1. How can we graph this quadratic inequality?
2. What would be the solution if the inequality was $3x^2 + 10x + 3 < 0$?
3. How do we solve inequalities with quadratic terms that don't factor easily?
4. What does the discriminant tell us about the nature of the solutions for quadratic inequalities?
5. How do we solve inequalities involving other types of polynomials (e.g., quartic)?

Tip:

To determine the sign of a quadratic function in different intervals, always check a point from each interval after finding the roots!