\[ P(x)=x^{4}+9 x^{3}+17 x^{2}-9 x-18 \] Write the polynomial in factored form. \( P(x)=\square \)

To factor the polynomial \( P(x) = x^{4} + 9x^{3} + 17x^{2} - 9x - 18 \), we can start by looking for rational roots using the Rational Root Theorem. After testing a few values, we find that \( x = -2 \) is a root. Using synthetic division, we can divide \( P(x) \) by \( x + 2 \), which gives us \( P(x) = (x + 2)(x^{3} + 7x^{2} + 7x - 9) \). Next, we can seek to factor the cubic polynomial \( x^{3} + 7x^{2} + 7x - 9 \). After trying more roots, we find that \( x = 1 \) is also a root. Dividing again, we eventually factor it down further, giving us: \[ P(x) = (x + 2)(x - 1)(x^2 + 8x + 9) \] The quadratic can be factored to give: \[ P(x) = (x + 2)(x - 1)(x + 3)(x + 3) \] So, we can write: \[ P(x) = (x + 2)(x - 1)(x + 3)^2 \]