Irritation from a cold or infection can cause a little blood to be coughed up in your phlegm. However it sounds like you have more blood than phlegm. Sometimes coughed up blood can come from the nose, rather than the lungs. When this happens there is usually a nosebleed also.
You need to have a thorough physical exam by your doctor. A chest x-ray may also be needed. Only then will the cause of the bright red blood and the best treatment be known.
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