Two polynomials \( P \) and \( D \) are given. Use either synthetic or long division to divide \( P(x) \) by \( D(x) \), and express the quotient \( P(x) / D(x) \) in the form \( \frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)} \). \[ P(x)=x^{5}+x^{4}-5 x^{3}+4 x+4, D(x)=x^{2}+x-1 \] \( D(x) \)\( \square \)

Given the polynomials \( P(x) = x^{5} + x^{4} - 5x^{3} + 4x + 4 \) and \( D(x) = x^{2} + x - 1 \), we need to perform polynomial division to find the quotient \( Q(x) \) and the remainder \( R(x) \) when \( P(x) \) is divided by \( D(x) \). To perform polynomial division, we can use synthetic division. The steps for synthetic division are as follows: 1. Write down the coefficients of the dividend polynomial \( P(x) \) in descending order of powers of \( x \). 2. Write down the divisor polynomial \( D(x) \) in the form of \( x - r \), where \( r \) is the root of the divisor polynomial. 3. Bring down the first coefficient of the dividend polynomial. 4. Multiply the root \( r \) by the coefficient brought down and write the result below the next coefficient. 5. Add the coefficients in the second column. 6. Repeat steps 4 and 5 until all coefficients have been processed. 7. The last coefficient in the third column is the remainder \( R(x) \). 8. The coefficients in the second column are the coefficients of the quotient polynomial \( Q(x) \). Let's perform synthetic division to find the quotient and remainder. Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\frac{\left(x^{5}+x^{4}-5x^{3}+4x+4\right)}{\left(x^{2}+x-1\right)}\) - step1: Calculate: \(x^{3}-4x+4+\frac{-4x+8}{x^{2}+x-1}\) The quotient \( Q(x) \) is \( x^{3} - 4x + 4 \) and the remainder \( R(x) \) is \( -4x + 8 \). Therefore, the quotient \( P(x) / D(x) \) can be expressed as \( Q(x) + \frac{R(x)}{D(x)} = x^{3} - 4x + 4 + \frac{-4x + 8}{x^{2} + x - 1} \).