So my HP Laptop has the "Intel (R) HD Graphis and the minimum requirement is a "NVIDIA GeForce 7/8/9/100/200 series / ATI RADEON X1950, HD 2000/3000/4000/5000 series" to run a certain game. Would it be possible to connect my laptop to an HD TV and run the game. I'm not sure if this is how video cards work.
Nope. Video card is a addon device which performs graphic related tasks and gives out signal to the monitor/tv to display.
With a Intel HD Graphics, the graphic related asks are performed by the processor which has very low performance compared to the video cards.
You cant possibly add video cards to your laptop. So the only way is to buy another laptop which has a video card in it.