Question

the number of positive real roots it has is equal to the number of sign changes or is less than that by an even number.

if $p(x)$ is a polynomial in standard form, then:
the number of negative real roots it has is equal to the number of sign changes in $p(-x)$ or is less than that by an even number.

use descartes rule of signs to find how many possible positive real roots this polynomial has:
$p(x)=4x^5 - 9x^4 -x^3 + 8x^2 + x$

3 or 1
1
2 or 0

Explanation:

Step1: List polynomial coefficients

$+4,\ -9,\ -1,\ +8,\ +1$

Step2: Count sign changes

1. $+4 \to -9$ (1st change)
2. $-1 \to +8$ (2nd change)

Total sign changes = 2

Step3: Apply Descartes' Rule

Possible positive roots = 2, or $2-2=0$

Answer:

2 or 0