Solve 4(x^3+4)^2+2(x^3+4)*3x*(4x-2)

$$4 ( x ^ { 3 } + 4 ) ^ { 2 } + 2 ( x ^ { 3 } + 4 ) \cdot 3 x \cdot ( 4 x - 2 )$$

Evaluate

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

Solution Steps

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

$$4\left(\left(x^{3}\right)^{2}+8x^{3}+16\right)+2\left(x^{3}+4\right)\times 3x\left(4x-2\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$4\left(x^{6}+8x^{3}+16\right)+2\left(x^{3}+4\right)\times 3x\left(4x-2\right)$$

Use the distributive property to multiply 4 by x^{6}+8x^{3}+16.

$$4x^{6}+32x^{3}+64+2\left(x^{3}+4\right)\times 3x\left(4x-2\right)$$

Multiply 2 and 3 to get 6.

$$4x^{6}+32x^{3}+64+6\left(x^{3}+4\right)x\left(4x-2\right)$$

Use the distributive property to multiply 6 by x^{3}+4.

$$4x^{6}+32x^{3}+64+\left(6x^{3}+24\right)x\left(4x-2\right)$$

Use the distributive property to multiply 6x^{3}+24 by x.

$$4x^{6}+32x^{3}+64+\left(6x^{4}+24x\right)\left(4x-2\right)$$

Use the distributive property to multiply 6x^{4}+24x by 4x-2.

$$4x^{6}+32x^{3}+64+24x^{5}-12x^{4}+96x^{2}-48x$$

Expand

$$4x^{6}+24x^{5}-12x^{4}+32x^{3}+96x^{2}-48x+64$$

Solution Steps

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

$$4\left(\left(x^{3}\right)^{2}+8x^{3}+16\right)+2\left(x^{3}+4\right)\times 3x\left(4x-2\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$4\left(x^{6}+8x^{3}+16\right)+2\left(x^{3}+4\right)\times 3x\left(4x-2\right)$$

Use the distributive property to multiply 4 by x^{6}+8x^{3}+16.

$$4x^{6}+32x^{3}+64+2\left(x^{3}+4\right)\times 3x\left(4x-2\right)$$

Multiply 2 and 3 to get 6.

$$4x^{6}+32x^{3}+64+6\left(x^{3}+4\right)x\left(4x-2\right)$$

Use the distributive property to multiply 6 by x^{3}+4.

$$4x^{6}+32x^{3}+64+\left(6x^{3}+24\right)x\left(4x-2\right)$$

Use the distributive property to multiply 6x^{3}+24 by x.

$$4x^{6}+32x^{3}+64+\left(6x^{4}+24x\right)\left(4x-2\right)$$

Use the distributive property to multiply 6x^{4}+24x by 4x-2.

$$4x^{6}+32x^{3}+64+24x^{5}-12x^{4}+96x^{2}-48x$$