We need to solve the inequality $x^2 + 4x - 3 > 0$.

1. Problem Description

We need to solve the inequality $x^2 + 4x - 3 > 0$.

2. Solution Steps

First, we find the roots of the quadratic equation $x^2 + 4x - 3 = 0$. We can use the quadratic formula:

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

In this case, $a=1$, $b=4$, and $c=-3$.

Plugging these values into the formula, we get:

$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-3)}}{2(1)}$

$x = \frac{-4 \pm \sqrt{16 + 12}}{2}$

$x = \frac{-4 \pm \sqrt{28}}{2}$

$x = \frac{-4 \pm 2\sqrt{7}}{2}$

$x = -2 \pm \sqrt{7}$

So, the roots are $x_1 = -2 - \sqrt{7}$ and $x_2 = -2 + \sqrt{7}$.

Since the coefficient of $x^2$ is positive, the parabola opens upwards. This means that the inequality $x^2 + 4x - 3 > 0$ is satisfied when $x$ is less than the smaller root or greater than the larger root.

Therefore, the solution to the inequality is $x < -2 - \sqrt{7}$ or $x > -2 + \sqrt{7}$.

3. Final Answer

$x < -2 - \sqrt{7}$ or $x > -2 + \sqrt{7}$