Solve 4(m+n)^2-4(m+n)(n-2)+(n-2)^2

$$4 ( m + n ) ^ { 2 } - 4 ( m + n ) ( n - 2 ) + ( n - 2 ) ^ { 2 }$$

Evaluate

$$\left(2m+n+2\right)^{2}$$

Solution Steps

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

$$4\left(m^{2}+2mn+n^{2}\right)-4\left(m+n\right)\left(n-2\right)+\left(n-2\right)^{2}$$

Use the distributive property to multiply 4 by m^{2}+2mn+n^{2}.

$$4m^{2}+8mn+4n^{2}-4\left(m+n\right)\left(n-2\right)+\left(n-2\right)^{2}$$

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

$$4m^{2}+8mn+4n^{2}-4\left(m+n\right)\left(n-2\right)+n^{2}-4n+4$$

Use the distributive property to multiply -4 by m+n.

$$4m^{2}+8mn+4n^{2}+\left(-4m-4n\right)\left(n-2\right)+n^{2}-4n+4$$

Use the distributive property to multiply -4m-4n by n-2.

$$4m^{2}+8mn+4n^{2}-4mn+8m-4n^{2}+8n+n^{2}-4n+4$$

Combine 8mn and -4mn to get 4mn.

$$4m^{2}+4mn+4n^{2}+8m-4n^{2}+8n+n^{2}-4n+4$$

Combine 4n^{2} and -4n^{2} to get 0.

$$4m^{2}+4mn+8m+8n+n^{2}-4n+4$$

Combine 8n and -4n to get 4n.

$$4m^{2}+4mn+8m+4n+n^{2}+4$$

Expand

$$4m^{2}+4mn+8m+n^{2}+4n+4$$

Solution Steps

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

$$4\left(m^{2}+2mn+n^{2}\right)-4\left(m+n\right)\left(n-2\right)+\left(n-2\right)^{2}$$

Use the distributive property to multiply 4 by m^{2}+2mn+n^{2}.

$$4m^{2}+8mn+4n^{2}-4\left(m+n\right)\left(n-2\right)+\left(n-2\right)^{2}$$

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

$$4m^{2}+8mn+4n^{2}-4\left(m+n\right)\left(n-2\right)+n^{2}-4n+4$$

Use the distributive property to multiply -4 by m+n.

$$4m^{2}+8mn+4n^{2}+\left(-4m-4n\right)\left(n-2\right)+n^{2}-4n+4$$

Use the distributive property to multiply -4m-4n by n-2.

$$4m^{2}+8mn+4n^{2}-4mn+8m-4n^{2}+8n+n^{2}-4n+4$$

Combine 8mn and -4mn to get 4mn.

$$4m^{2}+4mn+4n^{2}+8m-4n^{2}+8n+n^{2}-4n+4$$

Combine 4n^{2} and -4n^{2} to get 0.

$$4m^{2}+4mn+8m+8n+n^{2}-4n+4$$

Combine 8n and -4n to get 4n.

$$4m^{2}+4mn+8m+4n+n^{2}+4$$