Solve 5(left(x-4right)^3-2(x-4))-8(4left(6-3xright)^2-6(6-3x)+8)

$$5( { \left(x-4 \right) }^{ 3 } -2(x-4))-8(4 { \left(6-3x \right) }^{ 2 } -6(6-3x)+8)$$

Evaluate

$$5x^{3}-348x^{2}+1238x-1208$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-4\right)^{3}.

$$5\left(x^{3}-12x^{2}+48x-64-2\left(x-4\right)\right)-8\left(4\left(6-3x\right)^{2}-6\left(6-3x\right)+8\right)$$

Use the distributive property to multiply -2 by x-4.

$$5\left(x^{3}-12x^{2}+48x-64-2x+8\right)-8\left(4\left(6-3x\right)^{2}-6\left(6-3x\right)+8\right)$$

Combine 48x and -2x to get 46x.

$$5\left(x^{3}-12x^{2}+46x-64+8\right)-8\left(4\left(6-3x\right)^{2}-6\left(6-3x\right)+8\right)$$

Add -64 and 8 to get -56.

$$5\left(x^{3}-12x^{2}+46x-56\right)-8\left(4\left(6-3x\right)^{2}-6\left(6-3x\right)+8\right)$$

Use the distributive property to multiply 5 by x^{3}-12x^{2}+46x-56.

$$5x^{3}-60x^{2}+230x-280-8\left(4\left(6-3x\right)^{2}-6\left(6-3x\right)+8\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-3x\right)^{2}.

$$5x^{3}-60x^{2}+230x-280-8\left(4\left(36-36x+9x^{2}\right)-6\left(6-3x\right)+8\right)$$

Use the distributive property to multiply 4 by 36-36x+9x^{2}.

$$5x^{3}-60x^{2}+230x-280-8\left(144-144x+36x^{2}-6\left(6-3x\right)+8\right)$$

Use the distributive property to multiply -6 by 6-3x.

$$5x^{3}-60x^{2}+230x-280-8\left(144-144x+36x^{2}-36+18x+8\right)$$

Subtract 36 from 144 to get 108.

$$5x^{3}-60x^{2}+230x-280-8\left(108-144x+36x^{2}+18x+8\right)$$

Combine -144x and 18x to get -126x.

$$5x^{3}-60x^{2}+230x-280-8\left(108-126x+36x^{2}+8\right)$$

Add 108 and 8 to get 116.

$$5x^{3}-60x^{2}+230x-280-8\left(116-126x+36x^{2}\right)$$

Use the distributive property to multiply -8 by 116-126x+36x^{2}.

$$5x^{3}-60x^{2}+230x-280-928+1008x-288x^{2}$$

Subtract 928 from -280 to get -1208.

$$5x^{3}-60x^{2}+230x-1208+1008x-288x^{2}$$

Combine 230x and 1008x to get 1238x.

$$5x^{3}-60x^{2}+1238x-1208-288x^{2}$$

Combine -60x^{2} and -288x^{2} to get -348x^{2}.

$$5x^{3}-348x^{2}+1238x-1208$$

Expand

$$5x^{3}-348x^{2}+1238x-1208$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-4\right)^{3}.

$$5\left(x^{3}-12x^{2}+48x-64-2\left(x-4\right)\right)-8\left(4\left(6-3x\right)^{2}-6\left(6-3x\right)+8\right)$$

Use the distributive property to multiply -2 by x-4.

$$5\left(x^{3}-12x^{2}+48x-64-2x+8\right)-8\left(4\left(6-3x\right)^{2}-6\left(6-3x\right)+8\right)$$

Combine 48x and -2x to get 46x.

$$5\left(x^{3}-12x^{2}+46x-64+8\right)-8\left(4\left(6-3x\right)^{2}-6\left(6-3x\right)+8\right)$$

Add -64 and 8 to get -56.

$$5\left(x^{3}-12x^{2}+46x-56\right)-8\left(4\left(6-3x\right)^{2}-6\left(6-3x\right)+8\right)$$

Use the distributive property to multiply 5 by x^{3}-12x^{2}+46x-56.

$$5x^{3}-60x^{2}+230x-280-8\left(4\left(6-3x\right)^{2}-6\left(6-3x\right)+8\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-3x\right)^{2}.

$$5x^{3}-60x^{2}+230x-280-8\left(4\left(36-36x+9x^{2}\right)-6\left(6-3x\right)+8\right)$$

Use the distributive property to multiply 4 by 36-36x+9x^{2}.

$$5x^{3}-60x^{2}+230x-280-8\left(144-144x+36x^{2}-6\left(6-3x\right)+8\right)$$

Use the distributive property to multiply -6 by 6-3x.

$$5x^{3}-60x^{2}+230x-280-8\left(144-144x+36x^{2}-36+18x+8\right)$$

Subtract 36 from 144 to get 108.

$$5x^{3}-60x^{2}+230x-280-8\left(108-144x+36x^{2}+18x+8\right)$$

Combine -144x and 18x to get -126x.

$$5x^{3}-60x^{2}+230x-280-8\left(108-126x+36x^{2}+8\right)$$

Add 108 and 8 to get 116.

$$5x^{3}-60x^{2}+230x-280-8\left(116-126x+36x^{2}\right)$$

Use the distributive property to multiply -8 by 116-126x+36x^{2}.

$$5x^{3}-60x^{2}+230x-280-928+1008x-288x^{2}$$

Subtract 928 from -280 to get -1208.

$$5x^{3}-60x^{2}+230x-1208+1008x-288x^{2}$$

Combine 230x and 1008x to get 1238x.

$$5x^{3}-60x^{2}+1238x-1208-288x^{2}$$

Combine -60x^{2} and -288x^{2} to get -348x^{2}.

$$5x^{3}-348x^{2}+1238x-1208$$