To find the quadratic equation given the roots $3$ and $-4$, we can use the fact that a quadratic equation with roots $r_1$ and $r_2$ can be written as:

$(x - r_1)(x - r_2) = 0$

In this case, $r_1 = 3$ and $r_2 = -4$, so the equation becomes:

$(x - 3)(x + 4) = 0$

Now, expand the product:

$x^2 + 4x - 3x - 12 = 0$

Simplifying:

$x^2 + x - 12 = 0$

Thus, the quadratic equation is:

$x^2 + x - 12 = 0$

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How would the quadratic equation change if the roots were $5$ and $2$?
2. What is the general form of a quadratic equation?
3. How do you find the vertex of a quadratic equation?
4. Can you determine the axis of symmetry for $x^2 + x - 12 = 0$?
5. What role does the discriminant play in quadratic equations?

Tip: You can always check your quadratic equation by substituting the roots back into the equation to verify.