Use synthetic division to divide 3x3 - 5x2 + 3 by (x-3).

The divisor is $(x-3)$, so we use 3 for the division.
The polynomial $3x^3 - 5x^2 + 3$ is missing an $x$ term, so we write it as $3x^3 - 5x^2 + 0x + 3$. The coefficients are $3, -5, 0, 3$.

$$\begin{array}{c|cccc} 3 & 3 & -5 & 0 & 3 \\ & & & & \\ \hline & & & & \end{array}$$

Bring down the first coefficient, 3.
Multiply 3 by the divisor 3 to get 9. Add 9 to $-5$ to get 4.
Multiply 4 by the divisor 3 to get 12. Add 12 to 0 to get 12.
Multiply 12 by the divisor 3 to get 36. Add 36 to 3 to get 39.

$$\begin{array}{c|cccc} 3 & 3 & -5 & 0 & 3 \\ & & 9 & 12 & 36 \\ \hline & 3 & 4 & 12 & 39 \end{array}$$

The numbers in the bottom row, excluding the last one, are the coefficients of the quotient. Since the original polynomial was degree 3 and we divided by a degree 1 polynomial, the quotient will be degree 2.
The last number in the bottom row is the remainder.

The coefficients of the quotient are $3, 4, 12$.
The quotient is $3x^2 + 4x + 12$.
The remainder is 39.

The quotient is $\boxed{3x^2 + 4x + 12}$
The remainder is $\boxed{39}$