Solve 81(x+1)^2+90(x+1)(y+2)+25(y+2)^2

$$81 ( x + 1 ) ^ { 2 } + 90 ( x + 1 ) ( y + 2 ) + 25 ( y + 2 ) ^ { 2 }$$

Evaluate

$$\left(9x+5y+19\right)^{2}$$

Solution Steps

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

$$81\left(x^{2}+2x+1\right)+90\left(x+1\right)\left(y+2\right)+25\left(y+2\right)^{2}$$

Use the distributive property to multiply 81 by x^{2}+2x+1.

$$81x^{2}+162x+81+90\left(x+1\right)\left(y+2\right)+25\left(y+2\right)^{2}$$

Use the distributive property to multiply 90 by x+1.

$$81x^{2}+162x+81+\left(90x+90\right)\left(y+2\right)+25\left(y+2\right)^{2}$$

Use the distributive property to multiply 90x+90 by y+2.

$$81x^{2}+162x+81+90xy+180x+90y+180+25\left(y+2\right)^{2}$$

Combine 162x and 180x to get 342x.

$$81x^{2}+342x+81+90xy+90y+180+25\left(y+2\right)^{2}$$

Add 81 and 180 to get 261.

$$81x^{2}+342x+261+90xy+90y+25\left(y+2\right)^{2}$$

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

$$81x^{2}+342x+261+90xy+90y+25\left(y^{2}+4y+4\right)$$

Use the distributive property to multiply 25 by y^{2}+4y+4.

$$81x^{2}+342x+261+90xy+90y+25y^{2}+100y+100$$

Combine 90y and 100y to get 190y.

$$81x^{2}+342x+261+90xy+190y+25y^{2}+100$$

Add 261 and 100 to get 361.

$$81x^{2}+342x+361+90xy+190y+25y^{2}$$

Expand

$$81x^{2}+90xy+342x+25y^{2}+190y+361$$

Solution Steps

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

$$81\left(x^{2}+2x+1\right)+90\left(x+1\right)\left(y+2\right)+25\left(y+2\right)^{2}$$

Use the distributive property to multiply 81 by x^{2}+2x+1.

$$81x^{2}+162x+81+90\left(x+1\right)\left(y+2\right)+25\left(y+2\right)^{2}$$

Use the distributive property to multiply 90 by x+1.

$$81x^{2}+162x+81+\left(90x+90\right)\left(y+2\right)+25\left(y+2\right)^{2}$$

Use the distributive property to multiply 90x+90 by y+2.

$$81x^{2}+162x+81+90xy+180x+90y+180+25\left(y+2\right)^{2}$$

Combine 162x and 180x to get 342x.

$$81x^{2}+342x+81+90xy+90y+180+25\left(y+2\right)^{2}$$

Add 81 and 180 to get 261.

$$81x^{2}+342x+261+90xy+90y+25\left(y+2\right)^{2}$$

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

$$81x^{2}+342x+261+90xy+90y+25\left(y^{2}+4y+4\right)$$

Use the distributive property to multiply 25 by y^{2}+4y+4.

$$81x^{2}+342x+261+90xy+90y+25y^{2}+100y+100$$

Combine 90y and 100y to get 190y.

$$81x^{2}+342x+261+90xy+190y+25y^{2}+100$$

Add 261 and 100 to get 361.

$$81x^{2}+342x+361+90xy+190y+25y^{2}$$