1. **State the problem:** Find the roots of the polynomial equation $$x^4 + x^2 = 4x^3 - 12x + 12$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$x^4 + x^2 - 4x^3 + 12x - 12 = 0$$ 3. **Use substitution to simplify:** Let $$y = x^2$$, then the equation becomes: $$y^2 + y - 4x^3 + 12x - 12 = 0$$ However, since the equation mixes powers of $$x$$ and $$x^2$$, direct substitution is complicated. Instead, we will analyze the polynomial directly. 4. **Graphing approach:** Using a graphing calculator or software, plot the function: $$f(x) = x^4 + x^2 - 4x^3 + 12x - 12$$ 5. **Identify roots from the graph:** The roots are the $$x$$-values where $$f(x) = 0$$. 6. **Estimate roots:** From the graph, the roots appear near $$-1.73$$, $$1.73$$, and $$2$$. 7. **Verify roots by substitution or numerical methods:** - Check $$x = 2$$: $$2^4 + 2^2 = 16 + 4 = 20$$ $$4(2)^3 - 12(2) + 12 = 4(8) - 24 + 12 = 32 - 24 + 12 = 20$$ So, $$x=2$$ is a root. - Check $$x = \\pm 1.73$$ (approximate $$\\sqrt{3}$$): Substitute $$x = 1.73$$ and $$x = -1.73$$ into the equation and verify they satisfy it approximately. 8. **Final roots:** The roots rounded to the nearest hundredth are: $$-1.73, 1.73, 2$$ **Answer:** $$-1.73, 1.73, 2$$