Solve (3x+2)(x+y-2)-(3x+2)(x+4-7)(3x+2)

$$( 3 x + 2 ) ( x + y - 2 ) - ( 3 x + 2 ) ( x + 4 - 7 ) ( 3 x + 2 )$$

Evaluate

$$\left(3x+2\right)\left(4+y+8x-3x^{2}\right)$$

Solution Steps

Multiply 3x+2 and 3x+2 to get \left(3x+2\right)^{2}.

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

Apply the distributive property by multiplying each term of 3x+2 by each term of x+y-2.

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

Combine -6x and 2x to get -4x.

$$3x^{2}+3xy-4x+2y-4-\left(3x+2\right)^{2}\left(x+4-7\right)$$

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

$$3x^{2}+3xy-4x+2y-4-\left(9x^{2}+12x+4\right)\left(x+4-7\right)$$

Subtract 7 from 4 to get -3.

$$3x^{2}+3xy-4x+2y-4-\left(9x^{2}+12x+4\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 9x^{2}+12x+4 by each term of x-3.

$$3x^{2}+3xy-4x+2y-4-\left(9x^{3}-27x^{2}+12x^{2}-36x+4x-12\right)$$

Combine -27x^{2} and 12x^{2} to get -15x^{2}.

$$3x^{2}+3xy-4x+2y-4-\left(9x^{3}-15x^{2}-36x+4x-12\right)$$

Combine -36x and 4x to get -32x.

$$3x^{2}+3xy-4x+2y-4-\left(9x^{3}-15x^{2}-32x-12\right)$$

To find the opposite of 9x^{3}-15x^{2}-32x-12, find the opposite of each term.

$$3x^{2}+3xy-4x+2y-4-9x^{3}-\left(-15x^{2}\right)-\left(-32x\right)-\left(-12\right)$$

The opposite of -15x^{2} is 15x^{2}.

$$3x^{2}+3xy-4x+2y-4-9x^{3}+15x^{2}-\left(-32x\right)-\left(-12\right)$$

The opposite of -32x is 32x.

$$3x^{2}+3xy-4x+2y-4-9x^{3}+15x^{2}+32x-\left(-12\right)$$

The opposite of -12 is 12.

$$3x^{2}+3xy-4x+2y-4-9x^{3}+15x^{2}+32x+12$$

Combine 3x^{2} and 15x^{2} to get 18x^{2}.

$$18x^{2}+3xy-4x+2y-4-9x^{3}+32x+12$$

Combine -4x and 32x to get 28x.

$$18x^{2}+3xy+28x+2y-4-9x^{3}+12$$

Add -4 and 12 to get 8.

$$18x^{2}+3xy+28x+2y+8-9x^{3}$$

Expand

$$8+2y+28x+3xy+18x^{2}-9x^{3}$$

Solution Steps

Multiply 3x+2 and 3x+2 to get \left(3x+2\right)^{2}.

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

Apply the distributive property by multiplying each term of 3x+2 by each term of x+y-2.

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

Combine -6x and 2x to get -4x.

$$3x^{2}+3xy-4x+2y-4-\left(3x+2\right)^{2}\left(x+4-7\right)$$

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

$$3x^{2}+3xy-4x+2y-4-\left(9x^{2}+12x+4\right)\left(x+4-7\right)$$

Subtract 7 from 4 to get -3.

$$3x^{2}+3xy-4x+2y-4-\left(9x^{2}+12x+4\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 9x^{2}+12x+4 by each term of x-3.

$$3x^{2}+3xy-4x+2y-4-\left(9x^{3}-27x^{2}+12x^{2}-36x+4x-12\right)$$

Combine -27x^{2} and 12x^{2} to get -15x^{2}.

$$3x^{2}+3xy-4x+2y-4-\left(9x^{3}-15x^{2}-36x+4x-12\right)$$

Combine -36x and 4x to get -32x.

$$3x^{2}+3xy-4x+2y-4-\left(9x^{3}-15x^{2}-32x-12\right)$$

To find the opposite of 9x^{3}-15x^{2}-32x-12, find the opposite of each term.

$$3x^{2}+3xy-4x+2y-4-9x^{3}-\left(-15x^{2}\right)-\left(-32x\right)-\left(-12\right)$$

The opposite of -15x^{2} is 15x^{2}.

$$3x^{2}+3xy-4x+2y-4-9x^{3}+15x^{2}-\left(-32x\right)-\left(-12\right)$$

The opposite of -32x is 32x.

$$3x^{2}+3xy-4x+2y-4-9x^{3}+15x^{2}+32x-\left(-12\right)$$

The opposite of -12 is 12.

$$3x^{2}+3xy-4x+2y-4-9x^{3}+15x^{2}+32x+12$$

Combine 3x^{2} and 15x^{2} to get 18x^{2}.

$$18x^{2}+3xy-4x+2y-4-9x^{3}+32x+12$$

Combine -4x and 32x to get 28x.

$$18x^{2}+3xy+28x+2y-4-9x^{3}+12$$

Add -4 and 12 to get 8.

$$18x^{2}+3xy+28x+2y+8-9x^{3}$$