For each power of #x# in descending order, pick out pairs of terms - one from each trinomial - whose product results in that power of #x# and add them together:
#color(green)(x^5)#: #2x^2*3x^3 = color(blue)(6x^5)#
#color(green)(x^4)#: #(2x^2*-x^2) + (-3x*3x^3) = -2x^4-9x^4 = color(blue)(-11x^4)#
#color(green)(x^3)#: #(2x^2*sqrt(2)x) + (-3x*-x^2) + (2*3x^3)#
#= 2sqrt(2)x^3+3x^3+6x^3 = color(blue)((9+2sqrt(2))x^3)#
#color(green)(x^2)#: #(-3x*sqrt(2)x) + (2*-x^2) = -3sqrt(2)x^2-2x^2#
#= color(blue)(-(2+3sqrt(2))x^2)#
#color(gree)(x)#: #2*sqrt(2)x = color(blue)(2sqrt(2)x)#
#color(green)(1)#: #color(blue)(0)#
So:
#(2x^2-3x+2)(3x^3-x^2+sqrt(2)x)#
#=6x^5-11x^4+(9+2sqrt(2))x^3-(2+3sqrt(2))x^2+2sqrt(2)x#
Actually in practice, I would just work with the coefficients:
#color(green)(x^5)#: #2*3 = color(blue)(6)#
#color(green)(x^4)#: #(2*-1) + (-3*3) = -2-9 = color(blue)(-11)#
#color(green)(x^3)#: #(2*sqrt(2)) + (-3*-1) + (2*3) = 2sqrt(2)+3+6 = color(blue)(9+2sqrt(2))#
#color(green)(x^2)#: #(-3*sqrt(2)) + (2*-1) = -3sqrt(2)-2 = color(blue)(-(2+3sqrt(2)))#
#color(green)(x)#: #2*sqrt(2) = color(blue)(2sqrt(2))#
#color(green)(1)#: #color(blue)(0)#
...and I would just write out the sum as I went along:
#6x^5-11x^4+(9+2sqrt(2))x^3-(2+3sqrt(2))x^2+2sqrt(2)x#
