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 $$ P(x)=D(x) \cdot Q(x)+R(x) $$. $$ P(x)=4 x^{2}+3 x-1, \quad D(x)=x+5 $$ $$ P(x)=\square $$

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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.
$$ P(x) = (x + 5)(4x - 17) + 84 $$

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 \( P(x)=D(x) \cdot Q(x)+R(x) \). \[ P(x)=4 x^{2}+3 x-1, \quad D(x)=x+5 \] \( P(x)=\square \)

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