I was thinking someone else would answer this, I'm not positive I've done my work correctly.

I would start with the "foil" method. You would take each number in the first polynomial and multiply it by each number in the second polynomial.

Remember your rules of exponents, as well as watch the +/- signs.

2x^3(5x^3-2x^2-4x-2)= 10x⁶-4x⁵-8x⁴-4x³

5x^2(5x^3-2x^2-4x-2)= 25x⁵-10x⁴-20x³-10x²

-3x(5x^3-2x^2-4x-2)= -15x⁴+6x³+12x²+6x

1(5x^3-2x^2-4x-2)= 5x^3-2x^2-4x-2

Now you add them together and you will get= 10x⁶+21x⁵-33x⁴-13x³+2x-2

I got this far last night, I'm not sure how to continue beyond this point?

Luckily, this problem looks harder than it actually is, it will just take a lot of writing.
When multiplying long polynomials like this, it's often much easier to draw a rectangle, and write one polynomial along the top, and the other down the side, like this:
(Draw a rectangle. Cut it with 4 lines down, and 4 lines across to make 16 boxes)

           2x³   5x²   -3x      1    
  5x³ |        |        |       |       |
-2x²  |        |        |       |       |
   -4x |        |        |       |       |
     -2|        |         |       |       |

Now multiply the numbers on the sides one-by-one and fill in their boxes:

         2x³     5x²    -3x      1    
 5x³  |10x⁶ |25x⁵  |-15x⁴ |5x³ |
-2x² |-4x⁵  |-10x⁴ |         |      |
 -4x |-8x⁴  |         |         |       |
   -2|-4x³  |         |          |       |

After completing all the boxes, combine like terms. Add together all the x⁶'s, then all the x⁵'s, then all the x⁴'s and so on...
(like terms will be in diagonals)

10x⁶+21x⁵-33x⁴+....

This long expression will be the product. Hope this helps!