Use Descarte's rule of signs to discuss the possibilities for the roots of the equation. Do not solve the equation

-5r^(4)+6r^(3)+9r-15=0

To use Descartes' rule of signs to discuss the possibilities for the roots of the equation, we need to examine the signs of the coefficients in the equation.

The given equation is: -5r^4 + 6r^3 + 9r - 15 = 0

Step 1: Count the number of sign changes in the expression.

In the equation, we have the following sign changes from left to right: -5 (-), +6 (+), +9 (+), -15 (-). There are two sign changes in total.

Step 2: Now consider the equation with its terms in reverse order: -5r^4 - 6r^3 + 9r + 15 = 0

Again, count the number of sign changes: -5 (-), -6 (-), +9 (+), +15 (+). In this case, there are no sign changes or an even number of sign changes.

According to Descartes' rule of signs, the number of positive roots of the equation is either equal to the number of sign changes or less than that by an even number. In this case, there are two sign changes, so there are two or zero positive roots in the equation.

Similarly, if we count the number of sign changes in the equation with all terms in reverse order, we can determine the number of negative roots. Here, there are no sign changes or an even number, indicating that there are zero or two negative roots.

Therefore, based on Descartes' rule of signs, the possibilities for the roots of the equation -5r^4 + 6r^3 + 9r - 15 = 0 are that there can be either two positive roots and zero negative roots or zero positive roots and two negative roots.