Solve =(x)(2)(x^2+1)(2x)+(x^2+1)^2(1)

$$= ( x ) ( 2 ) ( x ^ { 2 } + 1 ) ( 2 x ) + ( x ^ { 2 } + 1 ) ^ { 2 } ( 1 )$$

Evaluate

$$\left(5x^{2}+1\right)\left(x^{2}+1\right)$$

Solution Steps

Multiply x and x to get x^{2}.

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

Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply x^{2}\times 4 by x^{2}+1.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+1\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply x^{4}+2x^{2}+1 by 1.

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

Combine 4x^{4} and x^{4} to get 5x^{4}.

$$5x^{4}+x^{2}\times 4+2x^{2}+1$$

Combine x^{2}\times 4 and 2x^{2} to get 6x^{2}.

$$5x^{4}+6x^{2}+1$$

Expand

$$5x^{4}+6x^{2}+1$$

Solution Steps

Multiply x and x to get x^{2}.

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

Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply x^{2}\times 4 by x^{2}+1.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+1\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply x^{4}+2x^{2}+1 by 1.

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

Combine 4x^{4} and x^{4} to get 5x^{4}.

$$5x^{4}+x^{2}\times 4+2x^{2}+1$$

Combine x^{2}\times 4 and 2x^{2} to get 6x^{2}.

$$5x^{4}+6x^{2}+1$$