Solve (3x+b)^2-2(x+b)(x-b)+3x(x+2b)-3b(4x-b)

$$( 3 x + b ) ^ { 2 } - 2 ( x + b ) ( x - b ) + 3 x ( x + 2 b ) - 3 b ( 4 x - b )$$

Evaluate

$$10x^{2}+6b^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(3x+b\right)^{2}.

$$9x^{2}+6xb+b^{2}-2\left(x+b\right)\left(x-b\right)+3x\left(x+2b\right)-3b\left(4x-b\right)$$

Use the distributive property to multiply 3x by x+2b.

$$9x^{2}+6xb+b^{2}-2\left(x+b\right)\left(x-b\right)+3x^{2}+6xb-3b\left(4x-b\right)$$

Use the distributive property to multiply -2 by x+b.

$$9x^{2}+6xb+b^{2}+\left(-2x-2b\right)\left(x-b\right)+3x^{2}+6xb-3b\left(4x-b\right)$$

Use the distributive property to multiply -2x-2b by x-b and combine like terms.

$$9x^{2}+6xb+b^{2}-2x^{2}+2b^{2}+3x^{2}+6xb-3b\left(4x-b\right)$$

Combine 9x^{2} and -2x^{2} to get 7x^{2}.

$$7x^{2}+6xb+b^{2}+2b^{2}+3x^{2}+6xb-3b\left(4x-b\right)$$

Combine b^{2} and 2b^{2} to get 3b^{2}.

$$7x^{2}+6xb+3b^{2}+3x^{2}+6xb-3b\left(4x-b\right)$$

Combine 7x^{2} and 3x^{2} to get 10x^{2}.

$$10x^{2}+6xb+3b^{2}+6xb-3b\left(4x-b\right)$$

Combine 6xb and 6xb to get 12xb.

$$10x^{2}+12xb+3b^{2}-3b\left(4x-b\right)$$

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

$$10x^{2}+12xb+3b^{2}-12bx+3b^{2}$$

Combine 12xb and -12bx to get 0.

$$10x^{2}+3b^{2}+3b^{2}$$

Combine 3b^{2} and 3b^{2} to get 6b^{2}.

$$10x^{2}+6b^{2}$$

Expand

$$10x^{2}+6b^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(3x+b\right)^{2}.

$$9x^{2}+6xb+b^{2}-2\left(x+b\right)\left(x-b\right)+3x\left(x+2b\right)-3b\left(4x-b\right)$$

Use the distributive property to multiply 3x by x+2b.

$$9x^{2}+6xb+b^{2}-2\left(x+b\right)\left(x-b\right)+3x^{2}+6xb-3b\left(4x-b\right)$$

Use the distributive property to multiply -2 by x+b.

$$9x^{2}+6xb+b^{2}+\left(-2x-2b\right)\left(x-b\right)+3x^{2}+6xb-3b\left(4x-b\right)$$

Use the distributive property to multiply -2x-2b by x-b and combine like terms.

$$9x^{2}+6xb+b^{2}-2x^{2}+2b^{2}+3x^{2}+6xb-3b\left(4x-b\right)$$

Combine 9x^{2} and -2x^{2} to get 7x^{2}.

$$7x^{2}+6xb+b^{2}+2b^{2}+3x^{2}+6xb-3b\left(4x-b\right)$$

Combine b^{2} and 2b^{2} to get 3b^{2}.

$$7x^{2}+6xb+3b^{2}+3x^{2}+6xb-3b\left(4x-b\right)$$

Combine 7x^{2} and 3x^{2} to get 10x^{2}.

$$10x^{2}+6xb+3b^{2}+6xb-3b\left(4x-b\right)$$

Combine 6xb and 6xb to get 12xb.

$$10x^{2}+12xb+3b^{2}-3b\left(4x-b\right)$$

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

$$10x^{2}+12xb+3b^{2}-12bx+3b^{2}$$

Combine 12xb and -12bx to get 0.

$$10x^{2}+3b^{2}+3b^{2}$$

Combine 3b^{2} and 3b^{2} to get 6b^{2}.

$$10x^{2}+6b^{2}$$