Solve 3(3x+1)^2(2x-1)+9(3x+1)(2x-1)^2

$$3 ( 3 x + 1 ) ^ { 2 } ( 2 x - 1 ) + 9 ( 3 x + 1 ) ( 2 x - 1 ) ^ { 2 }$$

Evaluate

$$162x^{3}-63x^{2}-21x+6$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+1\right)^{2}.

$$3\left(9x^{2}+6x+1\right)\left(2x-1\right)+9\left(3x+1\right)\left(2x-1\right)^{2}$$

Use the distributive property to multiply 3 by 9x^{2}+6x+1.

$$\left(27x^{2}+18x+3\right)\left(2x-1\right)+9\left(3x+1\right)\left(2x-1\right)^{2}$$

Use the distributive property to multiply 27x^{2}+18x+3 by 2x-1 and combine like terms.

$$54x^{3}+9x^{2}-12x-3+9\left(3x+1\right)\left(2x-1\right)^{2}$$

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

$$54x^{3}+9x^{2}-12x-3+9\left(3x+1\right)\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply 9 by 3x+1.

$$54x^{3}+9x^{2}-12x-3+\left(27x+9\right)\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply 27x+9 by 4x^{2}-4x+1 and combine like terms.

$$54x^{3}+9x^{2}-12x-3+108x^{3}-72x^{2}-9x+9$$

Combine 54x^{3} and 108x^{3} to get 162x^{3}.

$$162x^{3}+9x^{2}-12x-3-72x^{2}-9x+9$$

Combine 9x^{2} and -72x^{2} to get -63x^{2}.

$$162x^{3}-63x^{2}-12x-3-9x+9$$

Combine -12x and -9x to get -21x.

$$162x^{3}-63x^{2}-21x-3+9$$

Add -3 and 9 to get 6.

$$162x^{3}-63x^{2}-21x+6$$

Expand

$$162x^{3}-63x^{2}-21x+6$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+1\right)^{2}.

$$3\left(9x^{2}+6x+1\right)\left(2x-1\right)+9\left(3x+1\right)\left(2x-1\right)^{2}$$

Use the distributive property to multiply 3 by 9x^{2}+6x+1.

$$\left(27x^{2}+18x+3\right)\left(2x-1\right)+9\left(3x+1\right)\left(2x-1\right)^{2}$$

Use the distributive property to multiply 27x^{2}+18x+3 by 2x-1 and combine like terms.

$$54x^{3}+9x^{2}-12x-3+9\left(3x+1\right)\left(2x-1\right)^{2}$$

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

$$54x^{3}+9x^{2}-12x-3+9\left(3x+1\right)\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply 9 by 3x+1.

$$54x^{3}+9x^{2}-12x-3+\left(27x+9\right)\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply 27x+9 by 4x^{2}-4x+1 and combine like terms.

$$54x^{3}+9x^{2}-12x-3+108x^{3}-72x^{2}-9x+9$$

Combine 54x^{3} and 108x^{3} to get 162x^{3}.

$$162x^{3}+9x^{2}-12x-3-72x^{2}-9x+9$$

Combine 9x^{2} and -72x^{2} to get -63x^{2}.

$$162x^{3}-63x^{2}-12x-3-9x+9$$

Combine -12x and -9x to get -21x.

$$162x^{3}-63x^{2}-21x-3+9$$

Add -3 and 9 to get 6.

$$162x^{3}-63x^{2}-21x+6$$