Explanation:
You've got a good start.
The numerator is an order 3 polynomial, and the denominator is an order 1 polynomial. So the result will be an order 2 polynomial, so there will be 3 terms, plus a possible remainder.
There are 2 terms in the denominator.
So draw a grid 4 boxes wide and 2 boxes tall.
Write each term in the denominator (2x and -3) down the right side.
Write each term in the numerator (2x³, x², -16x, and 15) starting at the left side and continuing along the bottom.
Boxes diagonal of each other must add up to the numerator term. For example, the top left box is the only box in that diagonal, so that must be equal to 2x³.
Each box is the product of the term in the top row and the term in the right column. So if the first box is 2x³, and the term at the right is 2x, then the term at the top is x².
x² · 2x = 2x³
And if the term at the top is x², and the term at the lower right is -3, then the term in the bottom left box is -3x².
x² · -3 = -3x²
And since the boxes in that diagonal add up to x², the other box must be 4x².
-3x² + 4x² = x²
Continue this pattern until all boxes are filled.