Notice that y3 is a factor of all three terms. So factor that out:
4y7 - 12y5 + 16y3 = y3 (4y4 - 12y2 + 16)
4 is also a factor of all 3 terms, so factor that out too:
y3 (4y4 - 12y2 + 16) = 4y3 (y4 - 3y2 + 4)
The expression in parentheses can't be factored, so that's as far as you can get.