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