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

We want to divide P(x) = 4x² + 3x − 1 by D(x) = x + 5 and express it in the form   P(x) = D(x) · Q(x) + R(x). Step 1. Write down the division:   4x² + 3x − 1 divided by (x + 5). Step 2. Perform the division (here using long division): • Divide 4x² by x to get 4x. This is the first term of Q(x). • Multiply (x + 5) by 4x: 4x² + 20x. • Subtract this product from the original polynomial:   (4x² + 3x − 1) − (4x² + 20x) = (4x² − 4x²) + (3x − 20x) − 1 = −17x − 1. • Now, divide −17x by x to get −17. This is the next term of Q(x). • Multiply (x + 5) by −17: −17x − 85. • Subtract:   (−17x − 1) − (−17x − 85) = (−17x + 17x) + (−1 + 85) = 84. Step 3. Write the results: • Quotient Q(x) = 4x − 17. • Remainder R(x) = 84. Thus, we have:   P(x) = (x + 5)(4x − 17) + 84.