Short Answer 
The process of synthetic division involves arranging the polynomial in descending order, setting up the division with the divisor (x + 6) by replacing x with -6, and performing the calculations with coefficients. This results in a final quotient of 4x¬≤ Р27x + 162 after processing the coefficients through addition and multiplication.
Step 1: Arrange the Dividend
To begin, arrange the polynomial you are dividing, which is 4x¬≥ Р3x¬≤ + 5x + 6, in descending order of degree. Ensure all terms are present; if any degree is missing, use a coefficient of 0. For this example, the coefficients are (4, -3, 5, 6), which represent the polynomial terms.
Step 2: Set Up Synthetic Division
Next, set up the synthetic division format. Use the divisor, which is (x + 6), where you will replace x with -6 for synthetic division purposes. Write down the coefficients (4, -3, 5, 6) in a row, and draw a line beneath them to begin the calculation.
Step 3: Perform the Synthetic Division
Start the synthetic division process by bringing down the first coefficient (4). Multiply it by -6 and write the result under the next coefficient. Continue this process by adding down the columns until all coefficients are processed. The final row will give you (4, -27, 162), which corresponds to the quotient 4x¬≤ Р27x + 162.