Answer :

We begin with the polynomial

[tex]$$
P(x) = x^3 + 5x^2 + 9x + 45.
$$[/tex]

Step 1. Find a rational root

By testing simple values, we try [tex]$x=-5$[/tex]:

[tex]$$
P(-5) = (-5)^3 + 5(-5)^2 + 9(-5) + 45 = -125 + 125 - 45 + 45 = 0.
$$[/tex]

Since [tex]$P(-5)=0$[/tex], it follows that [tex]$x=-5$[/tex] is a root. Therefore, [tex]$(x+5)$[/tex] is a factor of [tex]$P(x)$[/tex].

Step 2. Factor the polynomial

Divide [tex]$P(x)$[/tex] by [tex]$x+5$[/tex] to find the quadratic factor. The division results in:

[tex]$$
P(x) = (x+5)(x^2+9).
$$[/tex]

Step 3. Solve for the roots

- The factor [tex]$x+5=0$[/tex] gives

[tex]$$
x = -5.
$$[/tex]

- The quadratic factor [tex]$x^2+9=0$[/tex] leads to

[tex]$$
x^2 = -9 \quad \Longrightarrow \quad x = \pm \sqrt{-9} = \pm 3i.
$$[/tex]

Final Answer

The factored form of the polynomial is:

[tex]$$
P(x) = (x+5)(x^2+9),
$$[/tex]

and the roots of the polynomial are:

[tex]$$
x = -5,\quad x = 3i,\quad x = -3i.
$$[/tex]

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