Step-by-step explanation:
You want synthetic division explained for division of (2x^4 -3x^3 -5x^2 +3x +8) by (x-2).
Comment
In general, synthetic division is long division without writing the variables. It takes advantage of the fact that subtracting the product of the quotient term and the divisor makes the leading term go to zero at each step. This makes it possible to write the entire division on one line, instead of several. (The long division of these polynomials is shown in the second attachment.)
Synthetic division is most easily accomplished as division of a polynomial by a monic binomial—a binomial of the form x -a. Both dividend and divisor can be divided by the leading coefficient of the divisor if it is not 1.
Procedure
The first attachment shows the tableau that is used during the synthetic division process. At top left is the opposite of the divisor constant. (This is the value that makes the divisor zero.) By using the opposite of the constant, later steps can involve addition, rather than subtraction, which simplifies the process and reduces errors.
To the right of the divisor zero, the coefficients of the polynomial are listed in decreasing order of degree. If a polynomial term is missing, then a 0 is used for its coefficient. Typically, the tableau is drawn with a vertical line between the divisor zero and the polynomial coefficients.
A horizontal line is drawn below the coefficients, with enough space to allow for partial products to be written between the coefficients and the horizontal line. The first (left-most) dividend polynomial coefficient is copied below the line.
From here, steps for succeeding columns of the tableau are the same. Working left-to-right, the value just written below the line is multiplied by the divisor zero, and that product is written above the line in the next column to the right. The coefficient is added to that, and the sum is written below the line.
The sum in the final (rightmost) column is the remainder from the division. The quotient from the division has the set of coefficients that are the numbers below the line to the left of the remainder.
Result
As shown in the attachment, the result of using synthetic division for the given polynomials is ...
2 1 -3 -3 2
This means the quotient is 2x³ +x² -3x -3 and the remainder is 2. That is, ...
(2x⁴-3x³-5x²+3x+8) ÷ (x-2) = 2x³ +x² -3x -3 + 2/(x-2)
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Additional comment
Synthetic division is really designed for division by monic binomials. It can be extended (with extreme care) to division by monic polynomials of higher degree than 1. What makes it work is the fact that the leading dividend term disappears at each step, so no accounting for the coefficients of the leading terms is required.
The same trick of using sums instead of differences could be applied to long division—again, with extreme care.