These strips are usually arranged in numerous parallel segments of a few series LEDs. The RGB strips can be a little different, but they work basically the same way. It looks like your strip is analog control and not digital (which would have little ICs in the strip to control individual LED color), so the specific color ground lines are all in parallel. You should be able to connect numerous strips in parallel - 12V to 12V, red to red, green to green, and blue to blue to blue. That is the same reason you can cut the strips into smaller segments if you want.
There is a limit to how much current can flow through the strip itself, which varies between manufacturers. Your other limit to how many strips can be connected together is how much power your supply / remote controller can handle. It should specify a limit in watts / current or total number of LEDs or parallel segments. That information should be on a sticker on the power supply or be listed in the manual.
From your product page:
Power 72 W
Which is 6A at 12V DC. That means your power supply/controller has to be able to handle at least 144 W (12A @ 12V DC) to connect two strips in parallel. Although, it is never a good idea to run something long term near its maximum ratings, so the supply should really be rated for something closer to 200 W.
If you are able to power multiple strips from your supply, it would be best to power them with a star topology so the current for each strip is separated. For example, connecting Christmas lights together end to end is daisy chaining, and the current for every light strand has to flow through the strands before it. Powering multiple light strands from one power strip is more of a star topology, with the current to each strip only flowing through itself (think of the power strip as your power supply).
Increasing the voltage by that much will significantly increase the LED current - much more than you might think.
Most of those strips have 3- LEDs in series per section. The forward voltage on a Green or Blue LED is about 3.2V. The Red LEDs are about 1.7V each.
The intended voltage for those strips is about 12V. Let's use that as a starting point.
The increase in current for the Green and Blue LEDs is going to be about:
@ 12V: 12V - (3 * 3.2V) = 2.4V across the current limit resistors
@ 14V: 14V - (3 * 3.2V) = 4.4V across the current limit resistors
@ 16V: 16V - (3 * 3.2V) = 6.4V across the current limit resistors
So: increasing the supply voltage from 12V to 14V is going to increase the LED current by (4.4 / 2.4 * 100%) = 183%
Increasing the supply voltage from 12V to 16V is going to increase the LED current by (6.4 / 2.4 * 100%) = 267%
The situation is a little better with the Red LEDs. Doing the same math as above results in the following voltages across the current limit resistors for the Red LEDs:
12V: 6.9V across the current limit resistor
14V: 8.9V across the resistor
16V: 10.9V across the resistor.
That results in a 129% increase in current if running at 14V; a 158% increase in current if running at 16V.
I strongly suspect that the current limit resistors are going to be really unhappy. You will most likely also notice that the Red LED didn't get as bright as the Green & Blue LEDs at the higher voltages.
Bottom line: run the strips at the manufacturer's maximum voltage and feed power from both ends if possible.
Best Answer
This is my room:
There's another 4m RGB LED strip on the opposite wall. Both LED strips and the custom LED lamps in the middle of the ceiling are connected to an ATX PSU ( this one, which is overkill by the way, but silent which is important )
My particular RGB LED strips power is about 10W per meter. So about 80W of RGB LEDs in my room. They operate at 12V so there is 6,66 Amps flowing through them. My ATX power supply is rated at 45.8 Amps in the +12V rail, that's 550W, much more than needed for my setup. The LED lamps in the middle of the ceiling are 150 Watts of high power Cree LEDs, also operating at 12V and from the same power supply. ATX power supplies tend to be powerful and cheap for projects like the one you're attempting.
You should power your LED strips only on one end, but you're better off dividing strips into separate segments. The copper traces that carry current in LED strips tend to be thin and offer relatively high resistance. When the RGB strips are full-on there is no noticeable effects in the lighting throught my 4 meter strips, yet when they are controlled to light really dimly, the eye gets quite sensitive to slight lighting differences and LED strips "tail" gets noticeably dimmer than the "head" closely connected to the PSU. I suggest you divide your installation into 4 different equally lengthed segments, all connected to the same PSU with "big diameter" wires. You can search online for nominal wire resistance per meter of a given diameter and calculate the difference in power that your "close to the PSU" LED strips will have compared to the "far away from the PSU" ones.
Let me make one extra comment you didn't ask for. If this RGB system is the only source of light you will have in your room, bare two things in mind:
1- RGB LEDs are inefficient compared to white LEDs, you won't get the same luminosity from 10W of RGB LEDs than from 10W of white LEDs.
2- Combining red, green and blue does give you white light, but a very bad quality one. There's a thing called Color Rendering Index. Things illuminated with "white" light coming from RGB sources will seem strange. In contrast, the white LEDs in the middle of my ceiling have a very high CRI, which makes the room feel like if it was being illuminated by natural sunlight.