1. **State the problem:** Divide the polynomial $29x^4 - 7x - 5x^2 + 7x^3 + 20$ by the binomial $5x - 7$. 2. **Arrange the dividend in descending powers of $x$:** $$29x^4 + 7x^3 - 5x^2 - 7x + 20$$ 3. **Use polynomial long division:** - Divide the leading term of the dividend $29x^4$ by the leading term of the divisor $5x$: $$\frac{29x^4}{5x} = \frac{29}{5}x^3$$ - Multiply the divisor by $\frac{29}{5}x^3$: $$\left(5x - 7\right) \times \frac{29}{5}x^3 = 29x^4 - \frac{203}{5}x^3$$ - Subtract this from the dividend: $$\left(29x^4 + 7x^3\right) - \left(29x^4 - \frac{203}{5}x^3\right) = 7x^3 + \frac{203}{5}x^3 = \frac{238}{5}x^3$$ 4. **Bring down the next term $-5x^2$ and continue:** - New expression: $\frac{238}{5}x^3 - 5x^2$ - Divide leading term by $5x$: $$\frac{\frac{238}{5}x^3}{5x} = \frac{238}{25}x^2$$ - Multiply divisor by $\frac{238}{25}x^2$: $$\left(5x - 7\right) \times \frac{238}{25}x^2 = \frac{1190}{25}x^3 - \frac{1666}{25}x^2 = \frac{238}{5}x^3 - \frac{1666}{25}x^2$$ - Subtract: $$\left(\frac{238}{5}x^3 - 5x^2\right) - \left(\frac{238}{5}x^3 - \frac{1666}{25}x^2\right) = -5x^2 + \frac{1666}{25}x^2 = \frac{541}{25}x^2$$ 5. **Bring down the next term $-7x$ and continue:** - New expression: $\frac{541}{25}x^2 - 7x$ - Divide leading term by $5x$: $$\frac{\frac{541}{25}x^2}{5x} = \frac{541}{125}x$$ - Multiply divisor by $\frac{541}{125}x$: $$\left(5x - 7\right) \times \frac{541}{125}x = \frac{2705}{125}x^2 - \frac{3787}{125}x$$ - Subtract: $$\left(\frac{541}{25}x^2 - 7x\right) - \left(\frac{2705}{125}x^2 - \frac{3787}{125}x\right) = \frac{541}{25}x^2 - \frac{2705}{125}x^2 - 7x + \frac{3787}{125}x = 0 + \frac{312}{125}x$$ 6. **Bring down the last term $+20$ and continue:** - New expression: $\frac{312}{125}x + 20$ - Divide leading term by $5x$: $$\frac{\frac{312}{125}x}{5x} = \frac{312}{625}$$ - Multiply divisor by $\frac{312}{625}$: $$\left(5x - 7\right) \times \frac{312}{625} = \frac{1560}{625}x - \frac{2184}{625}$$ - Subtract: $$\left(\frac{312}{125}x + 20\right) - \left(\frac{1560}{625}x - \frac{2184}{625}\right) = 0 + \left(20 + \frac{2184}{625}\right) = \frac{13484}{625}$$ 7. **Write the final answer:** $$\text{Quotient} = \frac{29}{5}x^3 + \frac{238}{25}x^2 + \frac{541}{125}x + \frac{312}{625}$$ $$\text{Remainder} = \frac{13484}{625}$$ So, $$\frac{29x^4 - 7x - 5x^2 + 7x^3 + 20}{5x - 7} = \frac{29}{5}x^3 + \frac{238}{25}x^2 + \frac{541}{125}x + \frac{312}{625} + \frac{\frac{13484}{625}}{5x - 7}$$