Solve [(x+2y)^2-(x+4)(3x-y)-5y^2]

$$[ ( x + 2 y ) ^ { 2 } - ( x + 4 ) ( 3 x - y ) - 5 y ^ { 2 } ]$$

Evaluate

$$4y-y^{2}-12x+5xy-2x^{2}$$

Solution Steps

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

$$x^{2}+4xy+4y^{2}-\left(x+4\right)\left(3x-y\right)-5y^{2}$$

Use the distributive property to multiply x+4 by 3x-y.

$$x^{2}+4xy+4y^{2}-\left(3x^{2}-xy+12x-4y\right)-5y^{2}$$

To find the opposite of 3x^{2}-xy+12x-4y, find the opposite of each term.

$$x^{2}+4xy+4y^{2}-3x^{2}+xy-12x+4y-5y^{2}$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}+4xy+4y^{2}+xy-12x+4y-5y^{2}$$

Combine 4xy and xy to get 5xy.

$$-2x^{2}+5xy+4y^{2}-12x+4y-5y^{2}$$

Combine 4y^{2} and -5y^{2} to get -y^{2}.

$$-2x^{2}+5xy-y^{2}-12x+4y$$

Expand

$$4y-y^{2}-12x+5xy-2x^{2}$$

Solution Steps

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

$$x^{2}+4xy+4y^{2}-\left(x+4\right)\left(3x-y\right)-5y^{2}$$

Use the distributive property to multiply x+4 by 3x-y.

$$x^{2}+4xy+4y^{2}-\left(3x^{2}-xy+12x-4y\right)-5y^{2}$$

To find the opposite of 3x^{2}-xy+12x-4y, find the opposite of each term.

$$x^{2}+4xy+4y^{2}-3x^{2}+xy-12x+4y-5y^{2}$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}+4xy+4y^{2}+xy-12x+4y-5y^{2}$$

Combine 4xy and xy to get 5xy.

$$-2x^{2}+5xy+4y^{2}-12x+4y-5y^{2}$$

Combine 4y^{2} and -5y^{2} to get -y^{2}.

$$-2x^{2}+5xy-y^{2}-12x+4y$$