To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.
#(color(red)(m^2) - color(red)(5m) + color(red)(4))(color(blue)(m^2) + color(blue)(7m) - color(blue)(3))# becomes:
#(color(red)(m^2) xx color(blue)(m^2)) + (color(red)(m^2) xx color(blue)(7m)) - (color(red)(m^2) xx color(blue)(3)) - (color(red)(5m) xx color(blue)(m^2)) - (color(red)(5m) xx color(blue)(7m)) + (color(red)(5m) xx color(blue)(3)) + (color(red)(4) xx color(blue)(m^2)) + (color(red)(4) xx color(blue)(7m)) - (color(red)(4) xx color(blue)(3))#
#m^4 + 7m^3 - 3m^2 - 5m^3 - 35m^2 + 15m + 4m^2 + 28m - 12#
We can now group and combine like terms:
#m^4 + 7m^3 - 5m^3 - 3m^2 - 35m^2 + 4m^2 + 15m + 28m - 12#
#m^4 + (7 - 5)m^3 + (-3 - 35 + 4)m^2 + (15 + 28)m - 12#
#m^4 + 2m^3 + (-34)m^2 + 43m - 12#
#m^4 + 2m^3 - 34m^2 + 43m - 12#
