Solve 6x(3x-1)^2+2x^2(1-3x)^2

$$6 x ( 3 x - 1 ) ^ { 2 } + 2 x ^ { 2 } ( 1 - 3 x ) ^ { 2 }$$

Evaluate

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

Solution Steps

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

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

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

$$54x^{3}-36x^{2}+6x+2x^{2}\left(1-3x\right)^{2}$$

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

$$54x^{3}-36x^{2}+6x+2x^{2}\left(1-6x+9x^{2}\right)$$

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

$$54x^{3}-36x^{2}+6x+2x^{2}-12x^{3}+18x^{4}$$

Combine -36x^{2} and 2x^{2} to get -34x^{2}.

$$54x^{3}-34x^{2}+6x-12x^{3}+18x^{4}$$

Combine 54x^{3} and -12x^{3} to get 42x^{3}.

$$42x^{3}-34x^{2}+6x+18x^{4}$$

Expand

$$18x^{4}+42x^{3}-34x^{2}+6x$$

Solution Steps

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

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

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

$$54x^{3}-36x^{2}+6x+2x^{2}\left(1-3x\right)^{2}$$

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

$$54x^{3}-36x^{2}+6x+2x^{2}\left(1-6x+9x^{2}\right)$$

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

$$54x^{3}-36x^{2}+6x+2x^{2}-12x^{3}+18x^{4}$$

Combine -36x^{2} and 2x^{2} to get -34x^{2}.

$$54x^{3}-34x^{2}+6x-12x^{3}+18x^{4}$$

Combine 54x^{3} and -12x^{3} to get 42x^{3}.

$$42x^{3}-34x^{2}+6x+18x^{4}$$