Solve f_2^3(x^2-4x+2)dx=-4/3

$$f _ { 2 } ^ { 3 } ( x ^ { 2 } - 4 x + 2 ) d x = - \frac { 4 } { 3 }$$

Solve for d

$$d=-\frac{4}{3x\left(x^{2}-4x+2\right)f_{2}^{3}}$$

$$x\neq 0\text{ and }x\neq \sqrt{2}+2\text{ and }x\neq 2-\sqrt{2}\text{ and }f_{2}\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply f_{2}^{3} by x^{2}-4x+2.

$$\left(f_{2}^{3}x^{2}-4f_{2}^{3}x+2f_{2}^{3}\right)dx=-\frac{4}{3}$$

Use the distributive property to multiply f_{2}^{3}x^{2}-4f_{2}^{3}x+2f_{2}^{3} by d.

$$\left(f_{2}^{3}x^{2}d-4f_{2}^{3}xd+2f_{2}^{3}d\right)x=-\frac{4}{3}$$

Use the distributive property to multiply f_{2}^{3}x^{2}d-4f_{2}^{3}xd+2f_{2}^{3}d by x.

$$f_{2}^{3}dx^{3}-4f_{2}^{3}dx^{2}+2f_{2}^{3}dx=-\frac{4}{3}$$

Combine all terms containing d.

$$\left(f_{2}^{3}x^{3}-4f_{2}^{3}x^{2}+2f_{2}^{3}x\right)d=-\frac{4}{3}$$

The equation is in standard form.

$$\left(f_{2}^{3}x^{3}-4x^{2}f_{2}^{3}+2xf_{2}^{3}\right)d=-\frac{4}{3}$$

Divide both sides by f_{2}^{3}x^{3}-4f_{2}^{3}x^{2}+2f_{2}^{3}x.

$$\frac{\left(f_{2}^{3}x^{3}-4x^{2}f_{2}^{3}+2xf_{2}^{3}\right)d}{f_{2}^{3}x^{3}-4x^{2}f_{2}^{3}+2xf_{2}^{3}}=-\frac{\frac{4}{3}}{f_{2}^{3}x^{3}-4x^{2}f_{2}^{3}+2xf_{2}^{3}}$$

Dividing by f_{2}^{3}x^{3}-4f_{2}^{3}x^{2}+2f_{2}^{3}x undoes the multiplication by f_{2}^{3}x^{3}-4f_{2}^{3}x^{2}+2f_{2}^{3}x.

$$d=-\frac{\frac{4}{3}}{f_{2}^{3}x^{3}-4x^{2}f_{2}^{3}+2xf_{2}^{3}}$$

Divide -\frac{4}{3} by f_{2}^{3}x^{3}-4f_{2}^{3}x^{2}+2f_{2}^{3}x.

$$d=-\frac{4}{3x\left(x^{2}-4x+2\right)f_{2}^{3}}$$

Solve for f_2

$$f_{2}=-\frac{3^{\frac{2}{3}}\sqrt[3]{\frac{4}{dx\left(x^{2}-4x+2\right)}}}{3}$$

$$x\neq 0\text{ and }x\neq \sqrt{2}+2\text{ and }x\neq 2-\sqrt{2}\text{ and }d\neq 0$$