Solve 2(x-3)(4x+7)^2+8(x-3)^3(4x+7)

$$2 ( x - 3 ) ( 4 x + 7 ) ^ { 2 } + 8 ( x - 3 ) ^ { 3 } ( 4 x + 7 )$$

Evaluate

$$2\left(x-3\right)\left(4x+7\right)\left(4x^{2}-20x+43\right)$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+7\right)^{2}.

$$2\left(x-3\right)\left(16x^{2}+56x+49\right)+8\left(x-3\right)^{3}\left(4x+7\right)$$

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

$$\left(2x-6\right)\left(16x^{2}+56x+49\right)+8\left(x-3\right)^{3}\left(4x+7\right)$$

Use the distributive property to multiply 2x-6 by 16x^{2}+56x+49 and combine like terms.

$$32x^{3}+16x^{2}-238x-294+8\left(x-3\right)^{3}\left(4x+7\right)$$

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

$$32x^{3}+16x^{2}-238x-294+8\left(x^{3}-9x^{2}+27x-27\right)\left(4x+7\right)$$

Use the distributive property to multiply 8 by x^{3}-9x^{2}+27x-27.

$$32x^{3}+16x^{2}-238x-294+\left(8x^{3}-72x^{2}+216x-216\right)\left(4x+7\right)$$

Use the distributive property to multiply 8x^{3}-72x^{2}+216x-216 by 4x+7 and combine like terms.

$$32x^{3}+16x^{2}-238x-294+32x^{4}-232x^{3}+360x^{2}+648x-1512$$

Combine 32x^{3} and -232x^{3} to get -200x^{3}.

$$-200x^{3}+16x^{2}-238x-294+32x^{4}+360x^{2}+648x-1512$$

Combine 16x^{2} and 360x^{2} to get 376x^{2}.

$$-200x^{3}+376x^{2}-238x-294+32x^{4}+648x-1512$$

Combine -238x and 648x to get 410x.

$$-200x^{3}+376x^{2}+410x-294+32x^{4}-1512$$

Subtract 1512 from -294 to get -1806.

$$-200x^{3}+376x^{2}+410x-1806+32x^{4}$$

Expand

$$32x^{4}-200x^{3}+376x^{2}+410x-1806$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+7\right)^{2}.

$$2\left(x-3\right)\left(16x^{2}+56x+49\right)+8\left(x-3\right)^{3}\left(4x+7\right)$$

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

$$\left(2x-6\right)\left(16x^{2}+56x+49\right)+8\left(x-3\right)^{3}\left(4x+7\right)$$

Use the distributive property to multiply 2x-6 by 16x^{2}+56x+49 and combine like terms.

$$32x^{3}+16x^{2}-238x-294+8\left(x-3\right)^{3}\left(4x+7\right)$$

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

$$32x^{3}+16x^{2}-238x-294+8\left(x^{3}-9x^{2}+27x-27\right)\left(4x+7\right)$$

Use the distributive property to multiply 8 by x^{3}-9x^{2}+27x-27.

$$32x^{3}+16x^{2}-238x-294+\left(8x^{3}-72x^{2}+216x-216\right)\left(4x+7\right)$$

Use the distributive property to multiply 8x^{3}-72x^{2}+216x-216 by 4x+7 and combine like terms.

$$32x^{3}+16x^{2}-238x-294+32x^{4}-232x^{3}+360x^{2}+648x-1512$$

Combine 32x^{3} and -232x^{3} to get -200x^{3}.

$$-200x^{3}+16x^{2}-238x-294+32x^{4}+360x^{2}+648x-1512$$

Combine 16x^{2} and 360x^{2} to get 376x^{2}.

$$-200x^{3}+376x^{2}-238x-294+32x^{4}+648x-1512$$

Combine -238x and 648x to get 410x.

$$-200x^{3}+376x^{2}+410x-294+32x^{4}-1512$$

Subtract 1512 from -294 to get -1806.

$$-200x^{3}+376x^{2}+410x-1806+32x^{4}$$