1. **State the problem:** Solve the cubic equation $$2x^3 + 2x^2 - 1040 = 0$$ for $x$. 2. **Write the equation:** $$2x^3 + 2x^2 - 1040 = 0$$ 3. **Divide both sides by 2 to simplify:** $$\cancel{2}x^3 + \cancel{2}x^2 - \cancel{2}520 = 0$$ which simplifies to $$x^3 + x^2 - 520 = 0$$ 4. **Try to find rational roots using the Rational Root Theorem:** Possible roots are factors of 520, such as $\pm1, \pm2, \pm4, \pm5, \pm8, \pm10, \pm13, \pm20, \pm26, \pm40, \pm52, \pm65, \pm104, \pm130, \pm260, \pm520$. 5. **Test $x=5$:** $$5^3 + 5^2 - 520 = 125 + 25 - 520 = 150 - 520 = -370 \neq 0$$ 6. **Test $x=8$:** $$8^3 + 8^2 - 520 = 512 + 64 - 520 = 576 - 520 = 56 \neq 0$$ 7. **Test $x=10$:** $$10^3 + 10^2 - 520 = 1000 + 100 - 520 = 1100 - 520 = 580 \neq 0$$ 8. **Test $x= -10$:** $$(-10)^3 + (-10)^2 - 520 = -1000 + 100 - 520 = -1420 \neq 0$$ 9. **Test $x= 4$:** $$4^3 + 4^2 - 520 = 64 + 16 - 520 = 80 - 520 = -440 \neq 0$$ 10. **Test $x= 13$:** $$13^3 + 13^2 - 520 = 2197 + 169 - 520 = 2366 \neq 0$$ 11. **Test $x= -8$:** $$(-8)^3 + (-8)^2 - 520 = -512 + 64 - 520 = -968 \neq 0$$ 12. **Test $x= 5$ again (already tested), try $x= 10$ (already tested), try $x= 4$ (already tested). Try $x= 2$:** $$2^3 + 2^2 - 520 = 8 + 4 - 520 = -508 \neq 0$$ 13. **Try $x= -5$:** $$(-5)^3 + (-5)^2 - 520 = -125 + 25 - 520 = -620 \neq 0$$ 14. **Try $x= 1$:** $$1 + 1 - 520 = -518 \neq 0$$ 15. **Try $x= -1$:** $$-1 + 1 - 520 = -520 \neq 0$$ 16. **Try $x= 20$:** $$20^3 + 20^2 - 520 = 8000 + 400 - 520 = 7880 \neq 0$$ 17. **Try $x= -4$:** $$-64 + 16 - 520 = -568 \neq 0$$ 18. **Try $x= 10$ (already tested), try $x= 8$ (already tested).** 19. Since no simple rational root found, use numerical methods or approximate root. 20. **Use the Intermediate Value Theorem:** - At $x=7$, $$7^3 + 7^2 - 520 = 343 + 49 - 520 = -128 < 0$$ - At $x=8$, $$512 + 64 - 520 = 56 > 0$$ So root lies between 7 and 8. 21. **Approximate root near 7.7:** $$7.7^3 + 7.7^2 - 520 \approx 456.53 + 59.29 - 520 = 15.82 > 0$$ 22. **Approximate root near 7.6:** $$7.6^3 + 7.6^2 - 520 \approx 438.98 + 57.76 - 520 = -23.26 < 0$$ 23. **Root is between 7.6 and 7.7, approximate as $x \approx 7.65$** **Final answer:** $$x \approx 7.65$$