Answer
$x^2+2x^2-1+\frac{1}{x-2}$
Work Step by Step
We have to perform the division
$$\dfrac{f(x)}{x-a}=\dfrac{x^3-5x+3}{x-2}.$$
We use synthetic division.
First we write the coefficients of $f$ in order of descending coefficients and write the value at which $f$ is being evaluated to the left.
Then we bring down the leading coefficient, multiply the leading coefficient by the $x$-value, write the product under the second coefficient, and add.
Then we multiply the previous sum by the $x$-value, write the product under the third coefficient, and add. Perform this for all the remaining coefficients. We get the value of $f$ at the given $x$-value as the final sum.
$$\dfrac{x^3-5x+3}{x-2}=x^2+2x^2-1+\dfrac{1}{x-2}.$$
The table built in the given solution is correct. What is wrong is the way it was read to give the result: the quotient has one degree less than $f$, therefore it would be $x^2+2x-1$, while the last number from the last line of the table represents the reminder. The correct result is:
$$x^2+2x^2-1+\dfrac{1}{x-2}.$$
