(15 + 5x^2 - 3x^3 - 9x ) : ( 5 - 3x )

To divide the polynomial (15 + 5x^2 - 3x^3 - 9x) by (5 - 3x), we can use long division.

First, we divide the highest degree term of the dividend by the highest degree term of the divisor. In this case, it is (-3x^3) divided by (-3x), which gives us x^2.

x^2
_________________________
5 - 3x | -3x^3 + 5x^2 - 9x + 15

Next, we multiply the divisor (5 - 3x) by the quotient (x^2) and subtract it from the dividend.

x^2
_________________________
5 - 3x | -3x^3 + 5x^2 - 9x + 15
-(-3x^3 + 3x^2)
_______________
2x^2 - 9x

Now, we bring down the next term (-9x) and repeat the process.

x^2 + 2x
_________________________
5 - 3x | -3x^3 + 5x^2 - 9x + 15
-(-3x^3 + 3x^2)
_______________
2x^2 - 9x
-(2x^2 - 6x)
__________
-3x + 15

Finally, we bring down the last term (15) and divide it by the divisor.

x^2 + 2x - 3
_________________________
5 - 3x | -3x^3 + 5x^2 - 9x + 15
-(-3x^3 + 3x^2)
_______________
2x^2 - 9x
-(2x^2 - 6x)
__________
-3x + 15
-(-3x + 9)
__________
6

Therefore, the result of the division is x^2 + 2x - 3 with a remainder of 6.