To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

#(color(red)(8v^2) - color(red)(5v) + color(red)(3))(color(blue)(2v^2) + color(blue)(5v) - color(blue)(2))# becomes:

#(color(red)(8v^2) xx color(blue)(2v^2)) + (color(red)(8v^2) xx color(blue)(5v)) - (color(red)(8v^2) xx color(blue)(2)) - (color(red)(5v) xx color(blue)(2v^2)) - (color(red)(5v) xx color(blue)(5v)) + (color(red)(5v) xx color(blue)(2)) + (color(red)(3) xx color(blue)(2v^2)) + (color(red)(3) xx color(blue)(5v)) - (color(red)(3) xx color(blue)(2))#

#16v^4 + 40v^3 - 16v^2 - 10v^3 - 25v^2 + 10v + 6v^2 + 15v - 6#

We can now group and combine like terms:

#16v^4 + 40v^3 - 10v^3 - 16v^2 - 25v^2 + 6v^2 + 10v + 15v - 6#

#16v^4 + (40 - 10)v^3 + (-16 - 25 + 6)v^2 + (10 + 15)v - 6#

#16v^4 + 30v^3 + (-35)v^2 + 25v - 6#

#16v^4 + 30v^3 - 35v^2 + 25v - 6#