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)(x) + color(red)(4))(color(blue)(2x^2) - color(blue)(x) + color(blue)(1))# becomes:
#(color(red)(x^2) xx color(blue)(2x^2)) - (color(red)(x^2) xx color(blue)(x)) + (color(red)(x^2) xx color(blue)(1)) + (color(red)(x) xx color(blue)(2x^2)) - (color(red)(x) xx color(blue)(x)) + (color(red)(x) xx color(blue)(1)) + (color(red)(4) xx color(blue)(2x^2)) - (color(red)(4) xx color(blue)(x)) + (color(red)(4) xx color(blue)(1))#
#2x^4 - x^3 + x^2 + 2x^3 - x^2 + x + 8x^2 - 4x + 4#
We can now group and combine like terms:
#2x^4 + 2x^3 - x^3 + 8x^2 + x^2 - x^2 + x - 4x + 4#
#2x^4 + 2x^3 - 1x^3 + 8x^2 + 1x^2 - 1x^2 + 1x - 4x + 4#
#2x^4 + (2 - 1)x^3 + (8 + 1 - 1)x^2 + (1 - 4)x + 4#
#2x^4 + 1x^3 + 8x^2 + (-3)x + 4#
#2x^4 + x^3 + 8x^2 - 3x + 4#
