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

To factor the polynomial   P(x) = x⁴ + 9x³ + 17x² – 9x – 18 follow these steps: 1. Find a root by testing small integers. Try x = 1:   P(1) = 1 + 9 + 17 – 9 – 18 = 0 Since P(1) = 0, (x – 1) is a factor. 2. Divide P(x) by (x – 1). Using synthetic division with coefficients 1, 9, 17, –9, –18:   • Bring down 1.   • Multiply 1 by 1 and add to 9 to get 10.   • Multiply 10 by 1 and add to 17 to get 27.   • Multiply 27 by 1 and add to –9 to get 18.   • Multiply 18 by 1 and add to –18 to get 0. The quotient is x³ + 10x² + 27x + 18. 3. Factor the cubic quotient: x³ + 10x² + 27x + 18. Test x = –1:   (–1)³ + 10(–1)² + 27(–1) + 18 = –1 + 10 – 27 + 18 = 0 So, (x + 1) is a factor. Using synthetic division for the cubic with root –1 (coefficients: 1, 10, 27, 18):   • Bring down 1.   • Multiply 1 by –1 and add to 10 to get 9.   • Multiply 9 by –1 and add to 27 to get 18.   • Multiply 18 by –1 and add to 18 to get 0. The quadratic quotient is x² + 9x + 18. 4. Factor the quadratic: x² + 9x + 18 factors as (x + 3)(x + 6) because 3 + 6 = 9 and 3 · 6 = 18. 5. Combine all factors:   P(x) = (x – 1)(x + 1)(x + 3)(x + 6). Thus, the factored form of the polynomial is:   P(x) = (x – 1)(x + 1)(x + 3)(x + 6).