To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.
#(color(red)(x^2) - color(red)(2x) + color(red)(3))(color(blue)(3x^2) + color(blue)(5x) - color(blue)(7))# becomes:
#(color(red)(x^2) xx color(blue)(3x^2)) + (color(red)(x^2) xx color(blue)(5x)) - (color(red)(x^2) xx color(blue)(7)) - (color(red)(2x) xx color(blue)(3x^2)) - (color(red)(2x) xx color(blue)(5x)) + (color(red)(2x) xx color(blue)(7)) + (color(red)(3) xx color(blue)(3x^2)) + (color(red)(3) xx color(blue)(5x)) - (color(red)(3) xx color(blue)(7))#
#3x^4 + 5x^3 - 7x^2 - 6x^3 - 10x^2 + 14x + 9x^2 + 15x - 21#
We can now group and combine like terms:
#3x^4 + 5x^3 - 6x^3 - 7x^2 - 10x^2 + 9x^2 + 14x + 15x - 21#
#3x^4 + (5 - 6)x^3 + (-7 - 10 + 9)x^2 + (14 + 15)x - 21#
#3x^4 + (-1)x^3 + (-8)x^2 + 29x - 21#
#3x^4 - 1x^3 - 8x^2 + 29x - 21#
#3x^4 - x^3 - 8x^2 + 29x - 21#
