Solve left(y^2+2right)^2-left(y^2-1right)^2

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

Evaluate

$$6y^{2}+3$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Combine 4y^{2} and 2y^{2} to get 6y^{2}.

$$6y^{2}+4-1$$

Subtract 1 from 4 to get 3.

$$6y^{2}+3$$

Expand

$$6y^{2}+3$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Combine 4y^{2} and 2y^{2} to get 6y^{2}.

$$6y^{2}+4-1$$

Subtract 1 from 4 to get 3.

$$6y^{2}+3$$