Solve (3x+2)^{3}-(2x+3)^3-(3x-2)^3+(2x-3)^3-(6x-5)^2

$$( 3 x + 2 ) ^ { 3 } - ( 2 x + 3 ) ^ { 3 } - ( 3 x - 2 ) ^ { 3 } + ( 2 x - 3 ) ^ { 3 } - ( 6 x - 5 ) ^ { 2 }$$

Evaluate

$$60x-63$$

Solution Steps

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

$$27x^{3}+54x^{2}+36x+8-\left(2x+3\right)^{3}-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

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

$$27x^{3}+54x^{2}+36x+8-\left(8x^{3}+36x^{2}+54x+27\right)-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

To find the opposite of 8x^{3}+36x^{2}+54x+27, find the opposite of each term.

$$27x^{3}+54x^{2}+36x+8-8x^{3}-36x^{2}-54x-27-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine 27x^{3} and -8x^{3} to get 19x^{3}.

$$19x^{3}+54x^{2}+36x+8-36x^{2}-54x-27-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine 54x^{2} and -36x^{2} to get 18x^{2}.

$$19x^{3}+18x^{2}+36x+8-54x-27-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine 36x and -54x to get -18x.

$$19x^{3}+18x^{2}-18x+8-27-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Subtract 27 from 8 to get -19.

$$19x^{3}+18x^{2}-18x-19-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(3x-2\right)^{3}.

$$19x^{3}+18x^{2}-18x-19-\left(27x^{3}-54x^{2}+36x-8\right)+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

To find the opposite of 27x^{3}-54x^{2}+36x-8, find the opposite of each term.

$$19x^{3}+18x^{2}-18x-19-27x^{3}+54x^{2}-36x+8+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine 19x^{3} and -27x^{3} to get -8x^{3}.

$$-8x^{3}+18x^{2}-18x-19+54x^{2}-36x+8+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine 18x^{2} and 54x^{2} to get 72x^{2}.

$$-8x^{3}+72x^{2}-18x-19-36x+8+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine -18x and -36x to get -54x.

$$-8x^{3}+72x^{2}-54x-19+8+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Add -19 and 8 to get -11.

$$-8x^{3}+72x^{2}-54x-11+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(2x-3\right)^{3}.

$$-8x^{3}+72x^{2}-54x-11+8x^{3}-36x^{2}+54x-27-\left(6x-5\right)^{2}$$

Combine -8x^{3} and 8x^{3} to get 0.

$$72x^{2}-54x-11-36x^{2}+54x-27-\left(6x-5\right)^{2}$$

Combine 72x^{2} and -36x^{2} to get 36x^{2}.

$$36x^{2}-54x-11+54x-27-\left(6x-5\right)^{2}$$

Combine -54x and 54x to get 0.

$$36x^{2}-11-27-\left(6x-5\right)^{2}$$

Subtract 27 from -11 to get -38.

$$36x^{2}-38-\left(6x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-5\right)^{2}.

$$36x^{2}-38-\left(36x^{2}-60x+25\right)$$

To find the opposite of 36x^{2}-60x+25, find the opposite of each term.

$$36x^{2}-38-36x^{2}+60x-25$$

Combine 36x^{2} and -36x^{2} to get 0.

$$-38+60x-25$$

Subtract 25 from -38 to get -63.

$$-63+60x$$

Expand

$$60x-63$$

Solution Steps

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

$$27x^{3}+54x^{2}+36x+8-\left(2x+3\right)^{3}-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

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

$$27x^{3}+54x^{2}+36x+8-\left(8x^{3}+36x^{2}+54x+27\right)-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

To find the opposite of 8x^{3}+36x^{2}+54x+27, find the opposite of each term.

$$27x^{3}+54x^{2}+36x+8-8x^{3}-36x^{2}-54x-27-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine 27x^{3} and -8x^{3} to get 19x^{3}.

$$19x^{3}+54x^{2}+36x+8-36x^{2}-54x-27-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine 54x^{2} and -36x^{2} to get 18x^{2}.

$$19x^{3}+18x^{2}+36x+8-54x-27-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine 36x and -54x to get -18x.

$$19x^{3}+18x^{2}-18x+8-27-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Subtract 27 from 8 to get -19.

$$19x^{3}+18x^{2}-18x-19-\left(3x-2\right)^{3}+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(3x-2\right)^{3}.

$$19x^{3}+18x^{2}-18x-19-\left(27x^{3}-54x^{2}+36x-8\right)+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

To find the opposite of 27x^{3}-54x^{2}+36x-8, find the opposite of each term.

$$19x^{3}+18x^{2}-18x-19-27x^{3}+54x^{2}-36x+8+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine 19x^{3} and -27x^{3} to get -8x^{3}.

$$-8x^{3}+18x^{2}-18x-19+54x^{2}-36x+8+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine 18x^{2} and 54x^{2} to get 72x^{2}.

$$-8x^{3}+72x^{2}-18x-19-36x+8+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Combine -18x and -36x to get -54x.

$$-8x^{3}+72x^{2}-54x-19+8+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Add -19 and 8 to get -11.

$$-8x^{3}+72x^{2}-54x-11+\left(2x-3\right)^{3}-\left(6x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(2x-3\right)^{3}.

$$-8x^{3}+72x^{2}-54x-11+8x^{3}-36x^{2}+54x-27-\left(6x-5\right)^{2}$$

Combine -8x^{3} and 8x^{3} to get 0.

$$72x^{2}-54x-11-36x^{2}+54x-27-\left(6x-5\right)^{2}$$

Combine 72x^{2} and -36x^{2} to get 36x^{2}.

$$36x^{2}-54x-11+54x-27-\left(6x-5\right)^{2}$$

Combine -54x and 54x to get 0.

$$36x^{2}-11-27-\left(6x-5\right)^{2}$$

Subtract 27 from -11 to get -38.

$$36x^{2}-38-\left(6x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-5\right)^{2}.

$$36x^{2}-38-\left(36x^{2}-60x+25\right)$$

To find the opposite of 36x^{2}-60x+25, find the opposite of each term.

$$36x^{2}-38-36x^{2}+60x-25$$

Combine 36x^{2} and -36x^{2} to get 0.

$$-38+60x-25$$

Subtract 25 from -38 to get -63.

$$-63+60x$$