Solve (v^2+1)^2-(v^2-2v-1)^2

$$( v ^ { 2 } + 1 ) ^ { 2 } - ( v ^ { 2 } - 2 v - 1 ) ^ { 2 }$$

Evaluate

$$4v\left(v^{2}-1\right)$$

Solution Steps

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

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

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

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

Square v^{2}-2v-1.

$$v^{4}+2v^{2}+1-\left(v^{4}-4v^{3}+2v^{2}+4v+1\right)$$

To find the opposite of v^{4}-4v^{3}+2v^{2}+4v+1, find the opposite of each term.

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

Combine v^{4} and -v^{4} to get 0.

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

Combine 2v^{2} and -2v^{2} to get 0.

$$1+4v^{3}-4v-1$$

Subtract 1 from 1 to get 0.

$$4v^{3}-4v$$

Expand

$$4v^{3}-4v$$

Solution Steps

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

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

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

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

Square v^{2}-2v-1.

$$v^{4}+2v^{2}+1-\left(v^{4}-4v^{3}+2v^{2}+4v+1\right)$$

To find the opposite of v^{4}-4v^{3}+2v^{2}+4v+1, find the opposite of each term.

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

Combine v^{4} and -v^{4} to get 0.

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

Combine 2v^{2} and -2v^{2} to get 0.

$$1+4v^{3}-4v-1$$

Subtract 1 from 1 to get 0.

$$4v^{3}-4v$$