Printing Quality dpi

I have a jpeg that is 1600x1200 pixels but only 72 dpi. I want to use it in a piece that is going to print. I’m certainly going to change the size of the image to 600x450 pixels but have it at print quality of 300dpi.

Will I still have poor quality because of the original 72 dpi?

Can you improve the print quality of an image that is very large at 72 dpi, by shrinking it down and then making 300dpi?