an astronaut must journey to a distant planet, which is 300 light years from earth. what speed will be necessary if the austronaut wishes to age only 12 years during the round trip

First, if you are inside a spaceship that travels with a relativistic speed, your inner time does not change. So your cells and DNA feel the time exactly as how they would feel it on Earth. This means that if you spend 12 years travelling on your spaceship, you will age those 12 years, while the ones that are still in Earth may be experienced time in a different way (The ones on Earth may measure the time between your departure and your arrival as 15 years, for example, so they feel the time in a different way).

Then we only need to solve the equation:

Distance/time = speed

Distance = 300 Light years.

Time = 12 years:

300 light-years/12 years  = speed = 25*C

where C is the speed of light:

C = 3*10^8 m/s

From this, we can conclude that you would need to travel a lot faster than the speed of light.

Then the speed is:

speed = 25*3*10^8 m/s= 7.5*10^9 m/s.