1. **State the problem:** We need to divide the polynomial $a^3 - 6a^2 + 10a - 3$ by the binomial $a - 3$. 2. **Formula and rule:** Polynomial division can be done using long division or synthetic division. Here, we use long division. 3. **Set up the division:** Divide the leading term of the dividend $a^3$ by the leading term of the divisor $a$ to get $a^2$. 4. **Multiply and subtract:** Multiply $a^2$ by the divisor $a - 3$ to get $a^3 - 3a^2$. Subtract this from the original polynomial: $$ (a^3 - 6a^2 + 10a - 3) - (a^3 - 3a^2) = \cancel{a^3} - 6a^2 + 10a - 3 - \cancel{a^3} + 3a^2 = -3a^2 + 10a - 3 $$ 5. **Repeat the process:** Divide the new leading term $-3a^2$ by $a$ to get $-3a$. 6. **Multiply and subtract:** Multiply $-3a$ by $a - 3$ to get $-3a^2 + 9a$. Subtract this from the current remainder: $$ (-3a^2 + 10a - 3) - (-3a^2 + 9a) = \cancel{-3a^2} + 10a - 3 - \cancel{-3a^2} - 9a = a - 3 $$ 7. **Continue:** Divide $a$ by $a$ to get $1$. 8. **Multiply and subtract:** Multiply $1$ by $a - 3$ to get $a - 3$. Subtract this from the remainder: $$ (a - 3) - (a - 3) = \cancel{a} - 3 - \cancel{a} + 3 = 0 $$ 9. **Conclusion:** The quotient is $a^2 - 3a + 1$ and the remainder is $0$. **Final answer:** $$ \frac{a^3 - 6a^2 + 10a - 3}{a - 3} = a^2 - 3a + 1 $$