Given \( P(x)=x^{3}+5 x^{2}+9 x+45 \). Write \( P \) in factored form (as a product of linear factors). Be sure to write the full equation, including \( P(x)= \).

We are given \[ P(x)=x^3+5x^2+9x+45. \] **Step 1. Find a Rational Root** To factor \(P(x)\) as a product of linear factors, we start by finding a rational root. By the Rational Root Theorem, any rational root is a divisor of the constant term (45). Possible candidates include \( \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45 \). We test \( x=-5 \): \[ P(-5)=(-5)^3+5(-5)^2+9(-5)+45=-125+125-45+45=0. \] Thus, \( x=-5 \) is a root and \( x+5 \) is a factor. **Step 2. Divide by \( (x+5) \)** Divide \(P(x)\) by \( x+5 \) to find the quadratic factor. Using synthetic division with root \( -5 \): \[ \begin{array}{r|rrrr} -5 & 1 & 5 & 9 & 45 \\ & & -5 & 0 & -45 \\ \hline & 1 & 0 & 9 & 0\\ \end{array} \] The quotient is \[ x^2+0x+9 = x^2+9. \] Thus, we have \[ P(x) = (x+5)(x^2+9). \] **Step 3. Factor the Quadratic Further** The quadratic \(x^2+9\) can be factored over the complex numbers. Recall that \[ x^2+9 = 0 \quad \Rightarrow \quad x^2 = -9 \quad \Rightarrow \quad x = \pm 3i. \] Thus, \[ x^2+9=(x+3i)(x-3i). \] **Step 4. Write the Fully Factored Form** Therefore, the complete factorization of \(P(x)\) into linear factors (over the complex numbers) is: \[ P(x) = (x+5)(x+3i)(x-3i). \]