Divide 3x3 - 5x2 + 3 by x-3 using synthetic division.

The divisor is $(x-3)$, so we use 3 for the division.
The coefficients of the polynomial $3x^3 - 5x^2 + 0x + 3$ are $3, -5, 0, 3$. (Note: A 0 is used for the missing $x$ term).

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

Bring down the first coefficient (3).
Multiply $3 \times 3 = 9$. Add to $-5$: $-5 + 9 = 4$.
Multiply $4 \times 3 = 12$. Add to 0: $0 + 12 = 12$.
Multiply $12 \times 3 = 36$. Add to 3: $3 + 36 = 39$.

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

The numbers in the bottom row, excluding the last one, are the coefficients of the quotient. Since the original polynomial was degree 3, the quotient will be degree 2.
The last number 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 $\boxed{39}$