A red giant is red because the inner core is white-hot, but as you get further away from the core, it cools and appears red hot. Once Carbon fusion starts, the force from fusion and gravity are equal and so the size and temperature of the red giant remain the same. Before moving into the next stage, the red giant is hot enough to fuse hydrogen into helium as well as helium into carbon.
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The core of the red giant runs out of helium to fuse into carbon, this causes the force of gravity to be greater than the force of fusion, which leads to the red giant contracting. The contraction forces the pressure to increase and with this the temperature.
When the Red giant finishes its stage of helium fusion, it ejects the layer of gas outside the core, which then becomes the planetary nebula. The core left behind is a white dwarf, called that because it is so hot it appears whitish blue.
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