**Proof:**

To show that the equation x^2 + ax - 4 = 0 has real and distinct roots for all real values of a, we can use the discriminant.

The discriminant is a mathematical term used to determine the nature of the roots of a quadratic equation. For a quadratic equation of the form ax^2 + bx + c = 0, the discriminant is given by the formula D = b^2 - 4ac.

In our given equation x^2 + ax - 4 = 0, the coefficients are:
a = 1
b = a
c = -4

**Calculating the Discriminant:**

Using the formula for the discriminant, we can calculate D as follows:

D = b^2 - 4ac

Substituting the values of a, b, and c into the formula, we get:

D = (a)^2 - 4(1)(-4)
D = a^2 + 16

**Analyzing the Discriminant:**

Now, let's analyze the discriminant D to determine the nature of the roots:

1. If D > 0, then the equation has two distinct real roots.
2. If D = 0, then the equation has one real root (a repeated root).
3. If D < 0,="" then="" the="" equation="" has="" no="" real="" roots="" (complex="" />

**Case 1: D > 0**

Since D = a^2 + 16, we can see that D is always positive for all real values of a. This implies that the equation x^2 + ax - 4 = 0 has two distinct real roots for all real values of a.

**Conclusion:**

Therefore, we have proved that the equation x^2 + ax - 4 = 0 has real and distinct roots for all real values of a.