Question

If a particlehas negative velocity and negative acceleration its speed is increases why and how

Answer 1

negative increasing velocity means it is going with in opposite direction with increase in speed

Answer 2

The sign of velocity and acceleration denotes it's direction only. The sign has nothing to do with the magnitude.

So when both acceleration and velocity are negative, both have the same direction or the direction of acceleration is along the direction of velocity. When this happens, the velocity increases in magnitude. You know that magnitude of velocity is called the speed. That's why speed increases.

If the direction of acceleration had been opposite of the velocity, then magnitude of velocity(speed) would have decreased.

When both acceleration and velocity have same direction (be it positive or negative), the magnitude of velocity increases.