Solve (x^2+sqrt{2x}+1)(x^2-sqrt{2x}+1)

$$( x ^ { 2 } + \sqrt { 2 x } + 1 ) ( x ^ { 2 } - \sqrt { 2 x } + 1 )$$

Evaluate

$$\left(x^{2}+1\right)^{2}-2x$$

Solution Steps

Use the distributive property to multiply x^{2}+\sqrt{2x}+1 by x^{2}-\sqrt{2x}+1 and combine like terms.

$$x^{4}+2x^{2}-\left(\sqrt{2x}\right)^{2}+1$$

Calculate \sqrt{2x} to the power of 2 and get 2x.

$$x^{4}+2x^{2}-2x+1$$

Differentiate w.r.t. x

$$4x^{3}+4x-2$$

Steps Using Derivative Rule for Sum

Use the distributive property to multiply x^{2}+\sqrt{2x}+1 by x^{2}-\sqrt{2x}+1 and combine like terms.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+2x^{2}-\left(\sqrt{2x}\right)^{2}+1)$$

Calculate \sqrt{2x} to the power of 2 and get 2x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+2x^{2}-2x+1)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$4x^{4-1}+2\times 2x^{2-1}-2x^{1-1}$$

Subtract 1 from 4.

$$4x^{3}+2\times 2x^{2-1}-2x^{1-1}$$

Multiply 2 times 2.

$$4x^{3}+4x^{2-1}-2x^{1-1}$$

Subtract 1 from 2.

$$4x^{3}+4x^{1}-2x^{1-1}$$

Subtract 1 from 1.

$$4x^{3}+4x^{1}-2x^{0}$$

For any term t, t^{1}=t.

$$4x^{3}+4x-2x^{0}$$

For any term t except 0, t^{0}=1.

$$4x^{3}+4x-2$$