Solve (12x-21)(2x+9)^2+(2x+9)^3(4)

$$( 12 x - 21 ) ( 2 x + 9 ) ^ { 2 } + ( 2 x + 9 ) ^ { 3 } ( 4 )$$

Evaluate

$$5\left(4x+3\right)\left(2x+9\right)^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+9\right)^{2}.

$$\left(12x-21\right)\left(4x^{2}+36x+81\right)+\left(2x+9\right)^{3}\times 4$$

Use the distributive property to multiply 12x-21 by 4x^{2}+36x+81 and combine like terms.

$$48x^{3}+348x^{2}+216x-1701+\left(2x+9\right)^{3}\times 4$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(2x+9\right)^{3}.

$$48x^{3}+348x^{2}+216x-1701+\left(8x^{3}+108x^{2}+486x+729\right)\times 4$$

Use the distributive property to multiply 8x^{3}+108x^{2}+486x+729 by 4.

$$48x^{3}+348x^{2}+216x-1701+32x^{3}+432x^{2}+1944x+2916$$

Combine 48x^{3} and 32x^{3} to get 80x^{3}.

$$80x^{3}+348x^{2}+216x-1701+432x^{2}+1944x+2916$$

Combine 348x^{2} and 432x^{2} to get 780x^{2}.

$$80x^{3}+780x^{2}+216x-1701+1944x+2916$$

Combine 216x and 1944x to get 2160x.

$$80x^{3}+780x^{2}+2160x-1701+2916$$

Add -1701 and 2916 to get 1215.

$$80x^{3}+780x^{2}+2160x+1215$$

Expand

$$80x^{3}+780x^{2}+2160x+1215$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+9\right)^{2}.

$$\left(12x-21\right)\left(4x^{2}+36x+81\right)+\left(2x+9\right)^{3}\times 4$$

Use the distributive property to multiply 12x-21 by 4x^{2}+36x+81 and combine like terms.

$$48x^{3}+348x^{2}+216x-1701+\left(2x+9\right)^{3}\times 4$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(2x+9\right)^{3}.

$$48x^{3}+348x^{2}+216x-1701+\left(8x^{3}+108x^{2}+486x+729\right)\times 4$$

Use the distributive property to multiply 8x^{3}+108x^{2}+486x+729 by 4.

$$48x^{3}+348x^{2}+216x-1701+32x^{3}+432x^{2}+1944x+2916$$

Combine 48x^{3} and 32x^{3} to get 80x^{3}.

$$80x^{3}+348x^{2}+216x-1701+432x^{2}+1944x+2916$$

Combine 348x^{2} and 432x^{2} to get 780x^{2}.

$$80x^{3}+780x^{2}+216x-1701+1944x+2916$$

Combine 216x and 1944x to get 2160x.

$$80x^{3}+780x^{2}+2160x-1701+2916$$

Add -1701 and 2916 to get 1215.

$$80x^{3}+780x^{2}+2160x+1215$$