1. **State the problem:** Solve the equation $$3x^4 - 4x^2 + 3x = -2$$ and explain why it has two solutions. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$3x^4 - 4x^2 + 3x + 2 = 0$$ 3. **Analyze the polynomial:** This is a quartic polynomial (degree 4). Quartic polynomials can have up to 4 real roots, but the actual number depends on the shape and factors. 4. **Attempt to factor the polynomial:** Try to factor by grouping or substitution. Let’s check for rational roots using the Rational Root Theorem. Possible roots are factors of 2 over factors of 3: $$\pm1, \pm2, \pm\frac{1}{3}, \pm\frac{2}{3}$$. 5. **Test possible roots:** - For $$x=1$$: $$3(1)^4 - 4(1)^2 + 3(1) + 2 = 3 - 4 + 3 + 2 = 4 \neq 0$$ - For $$x=-1$$: $$3(-1)^4 - 4(-1)^2 + 3(-1) + 2 = 3 - 4 - 3 + 2 = -2 \neq 0$$ - For $$x=2$$: $$3(16) - 4(4) + 3(2) + 2 = 48 - 16 + 6 + 2 = 40 \neq 0$$ - For $$x=-2$$: $$3(16) - 4(4) - 6 + 2 = 48 - 16 - 6 + 2 = 28 \neq 0$$ - For $$x=\frac{1}{3}$$: $$3\left(\frac{1}{3}\right)^4 - 4\left(\frac{1}{3}\right)^2 + 3\left(\frac{1}{3}\right) + 2 = 3\left(\frac{1}{81}\right) - 4\left(\frac{1}{9}\right) + 1 + 2 = \frac{3}{81} - \frac{4}{9} + 3 = \frac{1}{27} - \frac{4}{9} + 3 = \frac{1}{27} - \frac{12}{27} + \frac{81}{27} = \frac{70}{27} \neq 0$$ - For $$x=-\frac{1}{3}$$: $$3\left(\frac{1}{81}\right) - 4\left(\frac{1}{9}\right) - 1 + 2 = \frac{1}{27} - \frac{4}{9} + 1 = \frac{1}{27} - \frac{12}{27} + \frac{27}{27} = \frac{16}{27} \neq 0$$ No rational roots found. 6. **Use numerical methods or graphing:** The polynomial is continuous and quartic, so it can have 0, 2, or 4 real roots. By graphing or testing values, we find it crosses the x-axis twice. 7. **Approximate solutions:** Using numerical methods (e.g., Newton's method or graphing), the two real roots are approximately: $$x \approx -1.22$$ and $$x \approx 0.72$$ 8. **Why two solutions?** The polynomial's shape and sign changes indicate exactly two real roots. The other roots are complex. **Final answer:** The equation $$3x^4 - 4x^2 + 3x = -2$$ has exactly two real solutions approximately $$x \approx -1.22$$ and $$x \approx 0.72$$.