$-x^3-21x^2-135x-243$ is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.

The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction $\frac{p}{q}$, where p is a factor of the trailing constant and q is a factor of the leading coefficient.

The factor of the leading coefficient ( -1 ) is 1 .The factors of the constant term (-243) are 1 3 9 27 81 243 . Then the Rational Roots Tests yields the following possible solutions:

Substitute the POSSIBLE roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

If we plug these values into the polynomial $P(x)$, we obtain $P(-3)=0$.

To find remaining zeros we use Factor Theorem. This theorem states that if $\frac{p}{q}$ is root of the polynomial then this polynomial can be divided with $qx-p$. In this example:

Divide $P(x)$ with $x+3$

Polynomial $-x^2-18x-81$ can be used to find the remaining roots.

$-x^2-18x-81$ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.