Update: An aggression based car will work even better as you'll gain ALOT more boost when you hit vehicles and when you get boost bonuses from hitting every 10th car.
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In order to get a 10x multiplier in Showtime, you'll need to hit 10 buses after starting Showtime by pressing

and

together whilst free roaming around the city.
The road I used is called
Chubb Lane. Its located on west side of the map and has a gas station as well as repair shop on the same road (
Refer to 4:10 in the video for the specific location of the road. I show the gas station as a reference point.). These are irrelevant for the achievement but, I mention these as reference point for the achievement. This road is perfect for the achievement as there are 3 buses (
1:20 in the video) parked up on the side and stationary. You can easily hit these for the first 3 multipliers.
If you continue going south on
Chubb Lane, you'll eventually come up to an intersection (
2:00 in the video) where there will be another repair shop. Go left here on to
West Lake Drive and continue the showtime here. The reason for this is that there are 2 more buses (
3:04 in the video) parked up in a parking lot on the left hand side about halfway down this road, adding to your multiplier.
As you hit vehicles, your boost bar will fill up. Pressing the boost button (

) will give you some air and propel the car forward. For every 10 vehicles you hit, you'll get a significant amount of boost back in your boost bar, allowing you to go even further in the hopes of encountering more buses to add to your multiplier.
Couple of tips:
1. Stay in the oncoming lane and let vehicles comes to you
2. When you hit the ground and look as though you're about to stop, press

to use your boost to get some air and distance
3. Whilst in the air, you can use your left stick to control your movements (

,

,

and

)
4. You'll need a car with high boost capacity. I chose the
Jansen P12 which is a stunt car but, any stunt car will do
5. Vehicles tend to aim for you rather than avoid you, this makes hitting them easier
I've recorded and added a clip of me doing the showtime event on
Chubb Lane and getting to the magical 10x multiplier (Spoiler: I end up getting a 14x multipler by the end of the showtime). The achievement will unlock as soon as you hit the required multiplier and, you can either keep going on the showtime or, simply come to a stand still and let it end. Good luck.
If you leave a negative vote, please state why in the comments below so that I can improve the solution. Thanks.