1. **State the problem:** We need to find the quotient when dividing the polynomial $6x^3 - 43x^2 - 34x + 4$ by $6x + 5$. 2. **Recall the division formula:** Polynomial division is similar to long division with numbers. We divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by that result, subtract, and repeat. 3. **Divide the leading terms:** $$\frac{6x^3}{6x} = x^2$$ 4. **Multiply and subtract:** Multiply $6x + 5$ by $x^2$: $$x^2(6x + 5) = 6x^3 + 5x^2$$ Subtract this from the original polynomial: $$\left(6x^3 - 43x^2 - 34x + 4\right) - \left(6x^3 + 5x^2\right) = -48x^2 - 34x + 4$$ 5. **Repeat division with new polynomial:** Divide leading terms: $$\frac{-48x^2}{6x} = -8x$$ 6. **Multiply and subtract:** Multiply $6x + 5$ by $-8x$: $$-8x(6x + 5) = -48x^2 - 40x$$ Subtract: $$\left(-48x^2 - 34x + 4\right) - \left(-48x^2 - 40x\right) = 6x + 4$$ 7. **Repeat division:** Divide leading terms: $$\frac{6x}{6x} = 1$$ 8. **Multiply and subtract:** Multiply $6x + 5$ by $1$: $$6x + 5$$ Subtract: $$\left(6x + 4\right) - \left(6x + 5\right) = -1$$ 9. **Conclusion:** The quotient is the sum of the terms found: $$x^2 - 8x + 1$$ The remainder is $-1$, but since the problem asks only for the quotient, the answer is: $$\boxed{x^2 - 8x + 1}$$