1. Solve each equation by factoring. ### a) $x^4 - 16x^2 + 75 = 2x^2 - 6$ 1. Move all terms to one side: $$x^4 - 16x^2 + 75 - 2x^2 + 6 = 0$$ 2. Simplify: $$x^4 - 18x^2 + 81 = 0$$ 3. Recognize it as a quadratic in $x^2$: Let $y = x^2$, then $$y^2 - 18y + 81 = 0$$ 4. Factor the quadratic: $$(y - 9)^2 = 0$$ 5. So $y = 9$, thus $x^2 = 9$ 6. Solve for $x$: $$x = \pm 3$$ ### b) $2x^2 + 4x - 1 = x + 1$ 1. Move all terms to one side: $$2x^2 + 4x - 1 - x - 1 = 0$$ 2. Simplify: $$2x^2 + 3x - 2 = 0$$ 3. Factor the quadratic: $$(2x - 1)(x + 2) = 0$$ 4. Set each factor to zero: - $2x - 1 = 0 \Rightarrow x = \frac{1}{2}$ - $x + 2 = 0 \Rightarrow x = -2$ ### c) $4x^3 - x^2 - 2x + 2 = 3x^3 - 2(x^2 - 1)$ 1. Expand right side: $$3x^3 - 2x^2 + 2$$ 2. Move all terms to left side: $$4x^3 - x^2 - 2x + 2 - 3x^3 + 2x^2 - 2 = 0$$ 3. Simplify: $$x^3 + x^2 - 2x = 0$$ 4. Factor out $x$: $$x(x^2 + x - 2) = 0$$ 5. Factor quadratic: $$x(x + 2)(x - 1) = 0$$ 6. Solutions: $$x = 0, x = -2, x = 1$$ ### d) $-2x^2 + x - 6 = -x^3 + 2x - 8$ 1. Move all terms to one side: $$-2x^2 + x - 6 + x^3 - 2x + 8 = 0$$ 2. Simplify: $$x^3 - 2x^2 - x + 2 = 0$$ 3. Try factoring by grouping: $$x^3 - 2x^2 - x + 2 = (x^3 - 2x^2) - (x - 2)$$ 4. Factor each group: $$x^2(x - 2) - 1(x - 2)$$ 5. Factor out $(x - 2)$: $$(x - 2)(x^2 - 1) = 0$$ 6. Factor difference of squares: $$(x - 2)(x - 1)(x + 1) = 0$$ 7. Solutions: $$x = 2, x = 1, x = -1$$ Final answers: - a) $x = \pm 3$ - b) $x = \frac{1}{2}, -2$ - c) $x = 0, -2, 1$ - d) $x = 2, 1, -1$