Factor the polynomial completely. \[ -5 p^{3}+40 p^{2}+45 p \]

To factor the polynomial \(-5p^3 + 40p^2 + 45p\) completely, we can start by factoring out the greatest common factor (GCF) from all the terms. The GCF here is \(-5p\). Factoring that out gives us: \[ -5p(p^2 - 8p - 9) \] Next, we need to factor the quadratic \(p^2 - 8p - 9\). We look for two numbers that multiply to \(-9\) (the constant term) and add to \(-8\) (the coefficient of \(p\)). The numbers \(-9\) and \(1\) fit this requirement. Thus, we can factor the quadratic as follows: \[ p^2 - 8p - 9 = (p - 9)(p + 1) \] Putting it all together, we have: \[ -5p(p - 9)(p + 1) \] So, the completely factored form of the polynomial is: \[ -5p(p - 9)(p + 1) \]