Solve (r+1)^{2}r^{2}(2r+1)-r^2(r-1)^2(2r-1)=10r^4+2r^2.

$$( r + 1 ) ^ { 2 } r ^ { 2 } ( 2 r + 1 ) - r ^ { 2 } ( r - 1 ) ^ { 2 } ( 2 r - 1 ) = 10 r ^ { 4 } + 2 r ^ { 2 } .$$

Solve for r (complex solution)

$$r\in \mathrm{C}$$

Solution Steps

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

$$\left(r^{2}+2r+1\right)r^{2}\left(2r+1\right)-r^{2}\left(r-1\right)^{2}\left(2r-1\right)=10r^{4}+2r^{2}$$

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

$$\left(r^{4}+2r^{3}+r^{2}\right)\left(2r+1\right)-r^{2}\left(r-1\right)^{2}\left(2r-1\right)=10r^{4}+2r^{2}$$

Use the distributive property to multiply r^{4}+2r^{3}+r^{2} by 2r+1 and combine like terms.

$$2r^{5}+5r^{4}+4r^{3}+r^{2}-r^{2}\left(r-1\right)^{2}\left(2r-1\right)=10r^{4}+2r^{2}$$

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

$$2r^{5}+5r^{4}+4r^{3}+r^{2}-r^{2}\left(r^{2}-2r+1\right)\left(2r-1\right)=10r^{4}+2r^{2}$$

Use the distributive property to multiply r^{2} by r^{2}-2r+1.

$$2r^{5}+5r^{4}+4r^{3}+r^{2}-\left(r^{4}-2r^{3}+r^{2}\right)\left(2r-1\right)=10r^{4}+2r^{2}$$

Use the distributive property to multiply r^{4}-2r^{3}+r^{2} by 2r-1 and combine like terms.

$$2r^{5}+5r^{4}+4r^{3}+r^{2}-\left(2r^{5}-5r^{4}+4r^{3}-r^{2}\right)=10r^{4}+2r^{2}$$

To find the opposite of 2r^{5}-5r^{4}+4r^{3}-r^{2}, find the opposite of each term.

$$2r^{5}+5r^{4}+4r^{3}+r^{2}-2r^{5}+5r^{4}-4r^{3}+r^{2}=10r^{4}+2r^{2}$$

Combine 2r^{5} and -2r^{5} to get 0.

$$5r^{4}+4r^{3}+r^{2}+5r^{4}-4r^{3}+r^{2}=10r^{4}+2r^{2}$$

Combine 5r^{4} and 5r^{4} to get 10r^{4}.

$$10r^{4}+4r^{3}+r^{2}-4r^{3}+r^{2}=10r^{4}+2r^{2}$$

Combine 4r^{3} and -4r^{3} to get 0.

$$10r^{4}+r^{2}+r^{2}=10r^{4}+2r^{2}$$

Combine r^{2} and r^{2} to get 2r^{2}.

$$10r^{4}+2r^{2}=10r^{4}+2r^{2}$$

Subtract 10r^{4} from both sides.

$$10r^{4}+2r^{2}-10r^{4}=2r^{2}$$

Combine 10r^{4} and -10r^{4} to get 0.

$$2r^{2}=2r^{2}$$

Subtract 2r^{2} from both sides.

$$2r^{2}-2r^{2}=0$$

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

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any r.

$$r\in \mathrm{C}$$

Solve for r

$$r\in \mathrm{R}$$

Solution Steps

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

$$\left(r^{2}+2r+1\right)r^{2}\left(2r+1\right)-r^{2}\left(r-1\right)^{2}\left(2r-1\right)=10r^{4}+2r^{2}$$

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

$$\left(r^{4}+2r^{3}+r^{2}\right)\left(2r+1\right)-r^{2}\left(r-1\right)^{2}\left(2r-1\right)=10r^{4}+2r^{2}$$

Use the distributive property to multiply r^{4}+2r^{3}+r^{2} by 2r+1 and combine like terms.

$$2r^{5}+5r^{4}+4r^{3}+r^{2}-r^{2}\left(r-1\right)^{2}\left(2r-1\right)=10r^{4}+2r^{2}$$

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

$$2r^{5}+5r^{4}+4r^{3}+r^{2}-r^{2}\left(r^{2}-2r+1\right)\left(2r-1\right)=10r^{4}+2r^{2}$$

Use the distributive property to multiply r^{2} by r^{2}-2r+1.

$$2r^{5}+5r^{4}+4r^{3}+r^{2}-\left(r^{4}-2r^{3}+r^{2}\right)\left(2r-1\right)=10r^{4}+2r^{2}$$

Use the distributive property to multiply r^{4}-2r^{3}+r^{2} by 2r-1 and combine like terms.

$$2r^{5}+5r^{4}+4r^{3}+r^{2}-\left(2r^{5}-5r^{4}+4r^{3}-r^{2}\right)=10r^{4}+2r^{2}$$

To find the opposite of 2r^{5}-5r^{4}+4r^{3}-r^{2}, find the opposite of each term.

$$2r^{5}+5r^{4}+4r^{3}+r^{2}-2r^{5}+5r^{4}-4r^{3}+r^{2}=10r^{4}+2r^{2}$$

Combine 2r^{5} and -2r^{5} to get 0.

$$5r^{4}+4r^{3}+r^{2}+5r^{4}-4r^{3}+r^{2}=10r^{4}+2r^{2}$$

Combine 5r^{4} and 5r^{4} to get 10r^{4}.

$$10r^{4}+4r^{3}+r^{2}-4r^{3}+r^{2}=10r^{4}+2r^{2}$$

Combine 4r^{3} and -4r^{3} to get 0.

$$10r^{4}+r^{2}+r^{2}=10r^{4}+2r^{2}$$

Combine r^{2} and r^{2} to get 2r^{2}.

$$10r^{4}+2r^{2}=10r^{4}+2r^{2}$$

Subtract 10r^{4} from both sides.

$$10r^{4}+2r^{2}-10r^{4}=2r^{2}$$

Combine 10r^{4} and -10r^{4} to get 0.

$$2r^{2}=2r^{2}$$

Subtract 2r^{2} from both sides.

$$2r^{2}-2r^{2}=0$$

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

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any r.

$$r\in \mathrm{R}$$