1. **State the problem:** We need to express the polynomial $p(x)=6x^3-35x^2+34x+45$ in the form $p(x)=(2x-5)q(x)+r$, where $q(x)$ is a polynomial and $r$ is a constant. 2. **Formula and method:** This is polynomial division. We divide $p(x)$ by $2x-5$ to find quotient $q(x)$ and remainder $r$. 3. **Divide the leading terms:** Divide $6x^3$ by $2x$ to get $3x^2$. This is the first term of $q(x)$. 4. **Multiply and subtract:** Multiply $3x^2$ by $2x-5$ to get $6x^3 - 15x^2$. Subtract this from $p(x)$: $$ (6x^3 - 35x^2 + 34x + 45) - (6x^3 - 15x^2) = -20x^2 + 34x + 45 $$ 5. **Repeat division:** Divide $-20x^2$ by $2x$ to get $-10x$. This is the next term of $q(x)$. 6. **Multiply and subtract:** Multiply $-10x$ by $2x-5$ to get $-20x^2 + 50x$. Subtract: $$ (-20x^2 + 34x + 45) - (-20x^2 + 50x) = -16x + 45 $$ 7. **Repeat division:** Divide $-16x$ by $2x$ to get $-8$. This is the next term of $q(x)$. 8. **Multiply and subtract:** Multiply $-8$ by $2x-5$ to get $-16x + 40$. Subtract: $$ (-16x + 45) - (-16x + 40) = 5 $$ 9. **Remainder:** The remainder $r=5$ is a constant. 10. **Final answer:** $$ p(x) = (2x - 5)(3x^2 - 10x - 8) + 5 $$ This completes the division and expresses $p(x)$ in the desired form.