1. **State the problem:** We need to divide the polynomial $3x^2 - 18x - 46$ by the binomial $3x + 5$. 2. **Formula and method:** Polynomial division is similar to long division with numbers. We divide the leading term of the numerator by the leading term of the denominator, multiply the entire divisor by that result, subtract, and repeat. 3. **Step 1:** Divide the leading term $3x^2$ by $3x$ to get $x$. 4. **Step 2:** Multiply $3x + 5$ by $x$ to get $3x^2 + 5x$. 5. **Step 3:** Subtract this from the original polynomial: $$ (3x^2 - 18x - 46) - (3x^2 + 5x) = -23x - 46 $$ 6. **Step 4:** Divide the new leading term $-23x$ by $3x$ to get $-\frac{23}{3}$. 7. **Step 5:** Multiply $3x + 5$ by $-\frac{23}{3}$: $$ -\frac{23}{3} \times 3x = -23x $$ $$ -\frac{23}{3} \times 5 = -\frac{115}{3} $$ 8. **Step 6:** Subtract this from $-23x - 46$: $$ (-23x - 46) - (-23x - \frac{115}{3}) = -46 + \frac{115}{3} = -\frac{138}{3} + \frac{115}{3} = -\frac{23}{3} $$ 9. **Result:** The quotient is $$ x - \frac{23}{3} $$ and the remainder is $$ -\frac{23}{3} $$ 10. **Final answer:** $$ \frac{3x^2 - 18x - 46}{3x + 5} = x - \frac{23}{3} + \frac{-\frac{23}{3}}{3x + 5} $$ This means the division yields $x - \frac{23}{3}$ with a remainder of $-\frac{23}{3}$ over the divisor $3x + 5$.