1. **State the problem:** We want to divide the polynomial $$y = 2x^6 - 10x^5 - 5x^2 + 26x - 5$$ by the factor $$x - 5$$ using polynomial long division. 2. **Recall the formula and rules:** Polynomial long division is similar to numerical long division. We divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the divisor by that result, subtract, and repeat with the remainder. 3. **Set up the division:** Dividend: $$2x^6 - 10x^5 + 0x^4 + 0x^3 - 5x^2 + 26x - 5$$ (note the zero coefficients for missing powers) Divisor: $$x - 5$$ 4. **Step 1:** Divide $$2x^6$$ by $$x$$ to get $$2x^5$$. Multiply divisor by $$2x^5$$: $$2x^6 - 10x^5$$. Subtract: $$\left(2x^6 - 10x^5\right) - \left(2x^6 - 10x^5\right) = 0$$. Bring down the next term: $$0x^4$$. 5. **Step 2:** Divide $$0x^4$$ by $$x$$ to get $$0x^3$$. Multiply divisor by $$0x^3$$: $$0x^4 - 0x^3$$. Subtract: $$0x^4 - 0x^4 = 0$$. Bring down next term: $$0x^3$$. 6. **Step 3:** Divide $$0x^3$$ by $$x$$ to get $$0x^2$$. Multiply divisor by $$0x^2$$: $$0x^3 - 0x^2$$. Subtract: $$0x^3 - 0x^3 = 0$$. Bring down next term: $$-5x^2$$. 7. **Step 4:** Divide $$-5x^2$$ by $$x$$ to get $$-5x$$. Multiply divisor by $$-5x$$: $$-5x^2 + 25x$$. Subtract: $$(-5x^2 + 26x) - (-5x^2 + 25x) = 0x^2 + (26x - 25x) = x$$. Bring down next term: $$-5$$. 8. **Step 5:** Divide $$x$$ by $$x$$ to get $$1$$. Multiply divisor by $$1$$: $$x - 5$$. Subtract: $$(x - 5) - (x - 5) = 0$$. 9. **Conclusion:** The quotient is $$2x^5 + 0x^3 + 0x^2 - 5x + 1$$, which simplifies to $$2x^5 - 5x + 1$$. The remainder is $$0$$, so $$x - 5$$ is a factor. **Final answer:** $$\boxed{\frac{2x^6 - 10x^5 - 5x^2 + 26x - 5}{x - 5} = 2x^5 - 5x + 1}$$