We are tasked with solving the inequality:

\[
2x^2 + 4x - 3 < 0
\]

### Step 1: Solve the corresponding equation
First, solve the quadratic equation associated with the inequality:

\[
2x^2 + 4x - 3 = 0
\]

Using the quadratic formula:

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

Here, \(a = 2\), \(b = 4\), and \(c = -3\). Substituting into the formula:

\[
x = \frac{-4 \pm \sqrt{4^2 - 4(2)(-3)}}{2(2)}
\]
\[
x = \frac{-4 \pm \sqrt{16 + 24}}{4}
\]
\[
x = \frac{-4 \pm \sqrt{40}}{4}
\]
\[
x = \frac{-4 \pm 2\sqrt{10}}{4}
\]
\[
x = \frac{-2 \pm \sqrt{10}}{2}
\]

So, the roots are:

\[
x_1 = \frac{-2 + \sqrt{10}}{2}, \quad x_2 = \frac{-2 - \sqrt{10}}{2}
\]

### Step 2: Analyze the sign of the quadratic
The roots divide the number line into three intervals: 
\[
\left( -\infty, \frac{-2 - \sqrt{10}}{2} \right), \quad \left( \frac{-2 - \sqrt{10}}{2}, \frac{-2 + \sqrt{10}}{2} \right), \quad \left( \frac{-2 + \sqrt{10}}{2}, \infty \right)
\]

Next, test a value from each interval in the inequality \(2x^2 + 4x - 3 < 0\) to determine where the inequality holds.

1. For \(x = -2\) (from the middle interval):
   \[
   2(-2)^2 + 4(-2) - 3 = 8 - 8 - 3 = -3 \quad (\text{Negative, inequality holds here.})
   \]

2. For \(x = 0\) (from the right interval):
   \[
   2(0)^2 + 4(0) - 3 = -3 \quad (\text{Negative, inequality holds here.})
   \]

Hence, the solution to the inequality is the open interval:

\[
\boxed{\left( \frac{-2 - \sqrt{10}}{2}, \frac{-2 + \sqrt{10}}{2} \right)}
\]

Would you like a more detailed explanation or help with any further steps?

### Related Questions:
1. How would the inequality change if we had \(2x^2 + 4x - 3 \leq 0\)?
2. Can you explain how the quadratic formula derives from completing the square?
3. How do you analyze the behavior of quadratics geometrically (parabola properties)?
4. How does the discriminant help in determining the nature of the roots?
5. What happens if you change the inequality to \(2x^2 + 4x - 3 > 0\)?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) first—it tells you how many real roots the quadratic will have.

Master the steps to solve the quadratic inequality 2x² + 4x - 3 < 0. This solution involves using the quadratic formula and sign analysis to determine intervals where the inequality holds. Ideal for high school students in Grades 9-12.

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