1. **State the problem:** Perform long division on the product \((x-1)(x^3 - 70x^2 - 1203x - 1134)\). 2. **First, expand the product:** \[ (x-1)(x^3 - 70x^2 - 1203x - 1134) = x \cdot (x^3 - 70x^2 - 1203x - 1134) - 1 \cdot (x^3 - 70x^2 - 1203x - 1134) \] 3. Multiply each term: \[ x \cdot x^3 = x^4 \] \[ x \cdot (-70x^2) = -70x^3 \] \[ x \cdot (-1203x) = -1203x^2 \] \[ x \cdot (-1134) = -1134x \] \[ -1 \cdot x^3 = -x^3 \] \[ -1 \cdot (-70x^2) = +70x^2 \] \[ -1 \cdot (-1203x) = +1203x \] \[ -1 \cdot (-1134) = +1134 \] 4. **Combine like terms:** \[ x^4 + (-70x^3 - x^3) + (-1203x^2 + 70x^2) + (-1134x + 1203x) + 1134 \] \[ x^4 - 71x^3 - 1133x^2 + 69x + 1134 \] 5. **Now perform long division:** Divide \(x^4 - 71x^3 - 1133x^2 + 69x + 1134\) by \(x - 1\). 6. Divide the leading term \(x^4\) by \(x\) to get \(x^3\). 7. Multiply \(x^3\) by \(x - 1\) to get \(x^4 - x^3\). 8. Subtract \((x^4 - x^3)\) from \(x^4 - 71x^3\) to get \(-70x^3\). 9. Bring down the next term \(-1133x^2\), so the new expression is \(-70x^3 - 1133x^2\). 10. Divide \(-70x^3\) by \(x\) to get \(-70x^2\). 11. Multiply \(-70x^2\) by \(x - 1\) to get \(-70x^3 + 70x^2\). 12. Subtract \(-70x^3 + 70x^2\) from \(-70x^3 - 1133x^2\) to get \(-1203x^2\). 13. Bring down the next term \(+69x\), so the new expression is \(-1203x^2 + 69x\). 14. Divide \(-1203x^2\) by \(x\) to get \(-1203x\). 15. Multiply \(-1203x\) by \(x - 1\) to get \(-1203x^2 + 1203x\). 16. Subtract \(-1203x^2 + 1203x\) from \(-1203x^2 + 69x\) to get \(-1134x\). 17. Bring down the last term \(+1134\), so the new expression is \(-1134x + 1134\). 18. Divide \(-1134x\) by \(x\) to get \(-1134\). 19. Multiply \(-1134\) by \(x - 1\) to get \(-1134x + 1134\). 20. Subtract \(-1134x + 1134\) from \(-1134x + 1134\) to get remainder \(0\). 21. **Final answer:** The quotient is \(x^3 - 70x^2 - 1203x - 1134\) with remainder \(0\). This confirms that \((x-1)(x^3 - 70x^2 - 1203x - 1134)\) divided by \(x-1\) equals \(x^3 - 70x^2 - 1203x - 1134\).