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------------------------------------------------ Simplify the expression:

\[ x^3 - 5x^2 - 9x + 45 \]

Answer :

We start with the polynomial

$$
x^3 - 5x^2 - 9x + 45.
$$

**Step 1. Identify a Root**

A good strategy for factoring a cubic polynomial is to look for an integer root by testing simple values. By substituting $x = 5$, we see:

$$
5^3 - 5(5)^2 - 9(5) + 45 = 125 - 125 - 45 + 45 = 0.
$$

Since the polynomial equals zero, $x = 5$ is a root. This means that $(x - 5)$ is a factor of the polynomial.

**Step 2. Factor the Polynomial by Division**

To factor the polynomial further, divide it by $(x - 5)$. This division yields a quadratic polynomial:

$$
x^3 - 5x^2 - 9x + 45 = (x - 5)(x^2 - 9).
$$

**Step 3. Factor the Quadratic**

Notice that the quadratic

$$
x^2 - 9
$$

is a difference of squares. Recall that a difference of squares factors as

$$
a^2 - b^2 = (a - b)(a + b).
$$

Here, $a = x$ and $b = 3$, so

$$
x^2 - 9 = (x - 3)(x + 3).
$$

Thus, the complete factorization of the polynomial is:

$$
x^3 - 5x^2 - 9x + 45 = (x - 5)(x - 3)(x + 3).
$$

**Step 4. Find the Roots**

To find the roots of the polynomial, set each factor equal to zero:

1. From $x - 5 = 0$, we have $x = 5$.
2. From $x - 3 = 0$, we have $x = 3$.
3. From $x + 3 = 0$, we have $x = -3$.

The solutions (or roots) of the equation are:

$$
x = -3,\quad x = 3,\quad x = 5.
$$

**Final Answer**

The factorization of the polynomial is

$$
(x - 5)(x - 3)(x + 3),
$$

and the roots are $x = -3$, $3$, and $5$.