We are asked to divide the polynomial $45x^4 - 9x^3 + 27x^2$ by the monomial $-9x$.

1. Problem Description

We are asked to divide the polynomial $45x^4 - 9x^3 + 27x^2$ by the monomial $-9x$.

2. Solution Steps

To divide the polynomial by $-9x$, we divide each term of the polynomial by $-9x$.

$$\frac{45x^4 - 9x^3 + 27x^2}{-9x} = \frac{45x^4}{-9x} - \frac{9x^3}{-9x} + \frac{27x^2}{-9x}$$

Now, we simplify each term:

$\frac{45x^4}{-9x} = -5x^{4-1} = -5x^3$

$-\frac{9x^3}{-9x} = -(-1)x^{3-1} = x^2$

$\frac{27x^2}{-9x} = -3x^{2-1} = -3x$

Therefore,

$\frac{45x^4 - 9x^3 + 27x^2}{-9x} = -5x^3 + x^2 - 3x$

3. Final Answer

$-5x^3 + x^2 - 3x$