Solve 3(x-2)^2*x^2+(x-2)^3*2x

$$3 ( x - 2 ) ^ { 2 } \cdot x ^ { 2 } + ( x - 2 ) ^ { 3 } \cdot 2 x$$

Evaluate

$$x\left(5x-4\right)\left(x-2\right)^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 3 by x^{2}-4x+4.

$$\left(3x^{2}-12x+12\right)x^{2}+\left(x-2\right)^{3}\times 2x$$

Use the distributive property to multiply 3x^{2}-12x+12 by x^{2}.

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

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

$$3x^{4}-12x^{3}+12x^{2}+\left(x^{3}-6x^{2}+12x-8\right)\times 2x$$

Use the distributive property to multiply x^{3}-6x^{2}+12x-8 by 2.

$$3x^{4}-12x^{3}+12x^{2}+\left(2x^{3}-12x^{2}+24x-16\right)x$$

Use the distributive property to multiply 2x^{3}-12x^{2}+24x-16 by x.

$$3x^{4}-12x^{3}+12x^{2}+2x^{4}-12x^{3}+24x^{2}-16x$$

Combine 3x^{4} and 2x^{4} to get 5x^{4}.

$$5x^{4}-12x^{3}+12x^{2}-12x^{3}+24x^{2}-16x$$

Combine -12x^{3} and -12x^{3} to get -24x^{3}.

$$5x^{4}-24x^{3}+12x^{2}+24x^{2}-16x$$

Combine 12x^{2} and 24x^{2} to get 36x^{2}.

$$5x^{4}-24x^{3}+36x^{2}-16x$$

Expand

$$5x^{4}-24x^{3}+36x^{2}-16x$$

Solution Steps

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

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

Use the distributive property to multiply 3 by x^{2}-4x+4.

$$\left(3x^{2}-12x+12\right)x^{2}+\left(x-2\right)^{3}\times 2x$$

Use the distributive property to multiply 3x^{2}-12x+12 by x^{2}.

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

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

$$3x^{4}-12x^{3}+12x^{2}+\left(x^{3}-6x^{2}+12x-8\right)\times 2x$$

Use the distributive property to multiply x^{3}-6x^{2}+12x-8 by 2.

$$3x^{4}-12x^{3}+12x^{2}+\left(2x^{3}-12x^{2}+24x-16\right)x$$

Use the distributive property to multiply 2x^{3}-12x^{2}+24x-16 by x.

$$3x^{4}-12x^{3}+12x^{2}+2x^{4}-12x^{3}+24x^{2}-16x$$

Combine 3x^{4} and 2x^{4} to get 5x^{4}.

$$5x^{4}-12x^{3}+12x^{2}-12x^{3}+24x^{2}-16x$$

Combine -12x^{3} and -12x^{3} to get -24x^{3}.

$$5x^{4}-24x^{3}+12x^{2}+24x^{2}-16x$$

Combine 12x^{2} and 24x^{2} to get 36x^{2}.

$$5x^{4}-24x^{3}+36x^{2}-16x$$