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

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

Evaluate

$$5\left(x+1\right)\left(x+3\right)\left(x-2\right)^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by x+3.

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

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

$$2x^{4}-6x^{3}-12x^{2}+56x-48+\left(x+3\right)^{2}\times 3\left(x-2\right)^{2}$$

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

$$2x^{4}-6x^{3}-12x^{2}+56x-48+\left(x^{2}+6x+9\right)\times 3\left(x-2\right)^{2}$$

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

$$2x^{4}-6x^{3}-12x^{2}+56x-48+\left(x^{2}+6x+9\right)\times 3\left(x^{2}-4x+4\right)$$

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

$$2x^{4}-6x^{3}-12x^{2}+56x-48+\left(3x^{2}+18x+27\right)\left(x^{2}-4x+4\right)$$

Use the distributive property to multiply 3x^{2}+18x+27 by x^{2}-4x+4 and combine like terms.

$$2x^{4}-6x^{3}-12x^{2}+56x-48+3x^{4}+6x^{3}-33x^{2}-36x+108$$

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

$$5x^{4}-6x^{3}-12x^{2}+56x-48+6x^{3}-33x^{2}-36x+108$$

Combine -6x^{3} and 6x^{3} to get 0.

$$5x^{4}-12x^{2}+56x-48-33x^{2}-36x+108$$

Combine -12x^{2} and -33x^{2} to get -45x^{2}.

$$5x^{4}-45x^{2}+56x-48-36x+108$$

Combine 56x and -36x to get 20x.

$$5x^{4}-45x^{2}+20x-48+108$$

Add -48 and 108 to get 60.

$$5x^{4}-45x^{2}+20x+60$$

Expand

$$5x^{4}-45x^{2}+20x+60$$

Solution Steps

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

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

Use the distributive property to multiply 2 by x+3.

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

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

$$2x^{4}-6x^{3}-12x^{2}+56x-48+\left(x+3\right)^{2}\times 3\left(x-2\right)^{2}$$

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

$$2x^{4}-6x^{3}-12x^{2}+56x-48+\left(x^{2}+6x+9\right)\times 3\left(x-2\right)^{2}$$

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

$$2x^{4}-6x^{3}-12x^{2}+56x-48+\left(x^{2}+6x+9\right)\times 3\left(x^{2}-4x+4\right)$$

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

$$2x^{4}-6x^{3}-12x^{2}+56x-48+\left(3x^{2}+18x+27\right)\left(x^{2}-4x+4\right)$$

Use the distributive property to multiply 3x^{2}+18x+27 by x^{2}-4x+4 and combine like terms.

$$2x^{4}-6x^{3}-12x^{2}+56x-48+3x^{4}+6x^{3}-33x^{2}-36x+108$$

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

$$5x^{4}-6x^{3}-12x^{2}+56x-48+6x^{3}-33x^{2}-36x+108$$

Combine -6x^{3} and 6x^{3} to get 0.

$$5x^{4}-12x^{2}+56x-48-33x^{2}-36x+108$$

Combine -12x^{2} and -33x^{2} to get -45x^{2}.

$$5x^{4}-45x^{2}+56x-48-36x+108$$

Combine 56x and -36x to get 20x.

$$5x^{4}-45x^{2}+20x-48+108$$

Add -48 and 108 to get 60.

$$5x^{4}-45x^{2}+20x+60$$