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)(5x) - color(red)(1))(color(blue)(5x^2) - color(blue)(6x) + color(blue)(1))# becomes:
#(color(red)(x^2) xx color(blue)(5x^2)) - (color(red)(x^2) xx color(blue)(6x)) + (color(red)(x^2) xx color(blue)(1)) + (color(red)(5x) xx color(blue)(5x^2)) - (color(red)(5x) xx color(blue)(6x)) + (color(red)(5x) xx color(blue)(1)) - (color(red)(1) xx color(blue)(5x^2)) + (color(red)(1) xx color(blue)(6x)) - (color(red)(1) xx color(blue)(1))#
#5x^4 - 6x^3 + x^2 + 25x^3 - 30x^2 + 5x - 5x^2 + 6x - 1#
We can now group and combine like terms:
#5x^4 - 6x^3 + 25x^3 + x^2 - 30x^2 - 5x^2 + 5x + 6x - 1#
#5x^4 + (-6 + 25)x^3 + (1 - 30 - 5)x^2 + (5 + 6)x - 1#
#5x^4 + 19x^3 - 34x^2 + 11x - 1#
