Solve (x-5)^2(x^2+4)[(6x-5)(x)(2x)+(x^2+4)(3)(5)]

$$( x - 5 ) ^ { 2 } ( x ^ { 2 } + 4 ) [ ( 6 x - 5 ) ( x ) ( 2 x ) + ( x ^ { 2 } + 4 ) ( 3 ) ( 5 ) ]$$

Evaluate

$$\left(x-5\right)^{2}\left(x^{2}+4\right)\left(12x^{3}+5x^{2}+60\right)$$

Solution Steps

Multiply x and x to get x^{2}.

$$\left(x-5\right)^{2}\left(x^{2}+4\right)\left(\left(6x-5\right)x^{2}\times 2+\left(x^{2}+4\right)\times 3\times 5\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(\left(6x-5\right)x^{2}\times 2+\left(x^{2}+4\right)\times 3\times 5\right)$$

Use the distributive property to multiply 6x-5 by x^{2}.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(\left(6x^{3}-5x^{2}\right)\times 2+\left(x^{2}+4\right)\times 3\times 5\right)$$

Use the distributive property to multiply 6x^{3}-5x^{2} by 2.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(12x^{3}-10x^{2}+\left(x^{2}+4\right)\times 3\times 5\right)$$

Multiply 3 and 5 to get 15.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(12x^{3}-10x^{2}+\left(x^{2}+4\right)\times 15\right)$$

Use the distributive property to multiply x^{2}+4 by 15.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(12x^{3}-10x^{2}+15x^{2}+60\right)$$

Combine -10x^{2} and 15x^{2} to get 5x^{2}.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(12x^{3}+5x^{2}+60\right)$$

Use the distributive property to multiply x^{2}-10x+25 by x^{2}+4 and combine like terms.

$$\left(x^{4}+29x^{2}-10x^{3}-40x+100\right)\left(12x^{3}+5x^{2}+60\right)$$

Use the distributive property to multiply x^{4}+29x^{2}-10x^{3}-40x+100 by 12x^{3}+5x^{2}+60 and combine like terms.

$$12x^{7}-115x^{6}-275x^{4}+298x^{5}+2240x^{2}+400x^{3}-2400x+6000$$

Expand

$$12x^{7}-115x^{6}+298x^{5}-275x^{4}+400x^{3}+2240x^{2}-2400x+6000$$

Solution Steps

Multiply x and x to get x^{2}.

$$\left(x-5\right)^{2}\left(x^{2}+4\right)\left(\left(6x-5\right)x^{2}\times 2+\left(x^{2}+4\right)\times 3\times 5\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(\left(6x-5\right)x^{2}\times 2+\left(x^{2}+4\right)\times 3\times 5\right)$$

Use the distributive property to multiply 6x-5 by x^{2}.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(\left(6x^{3}-5x^{2}\right)\times 2+\left(x^{2}+4\right)\times 3\times 5\right)$$

Use the distributive property to multiply 6x^{3}-5x^{2} by 2.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(12x^{3}-10x^{2}+\left(x^{2}+4\right)\times 3\times 5\right)$$

Multiply 3 and 5 to get 15.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(12x^{3}-10x^{2}+\left(x^{2}+4\right)\times 15\right)$$

Use the distributive property to multiply x^{2}+4 by 15.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(12x^{3}-10x^{2}+15x^{2}+60\right)$$

Combine -10x^{2} and 15x^{2} to get 5x^{2}.

$$\left(x^{2}-10x+25\right)\left(x^{2}+4\right)\left(12x^{3}+5x^{2}+60\right)$$

Use the distributive property to multiply x^{2}-10x+25 by x^{2}+4 and combine like terms.

$$\left(x^{4}+29x^{2}-10x^{3}-40x+100\right)\left(12x^{3}+5x^{2}+60\right)$$

Use the distributive property to multiply x^{4}+29x^{2}-10x^{3}-40x+100 by 12x^{3}+5x^{2}+60 and combine like terms.

$$12x^{7}-115x^{6}-275x^{4}+298x^{5}+2240x^{2}+400x^{3}-2400x+6000$$