Solve left(x+1right)^3-left(x-2right)^3/9=left(2x^-5right)^2+11/4

$$\frac{ { \left(x+1 \right) }^{ 3 } - { \left(x-2 \right) }^{ 3 } }{ 9 } = \frac{ { \left(2 { x }^{ } -5 \right) }^{ 2 } +11 }{ 4 }$$

Solve for x

$$x=2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 36, the least common multiple of 9,4.

$$4\left(\left(x+1\right)^{3}-\left(x-2\right)^{3}\right)=9\left(\left(2x^{1}-5\right)^{2}+11\right)$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+1\right)^{3}.

$$4\left(x^{3}+3x^{2}+3x+1-\left(x-2\right)^{3}\right)=9\left(\left(2x^{1}-5\right)^{2}+11\right)$$

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

$$4\left(x^{3}+3x^{2}+3x+1-\left(x^{3}-6x^{2}+12x-8\right)\right)=9\left(\left(2x^{1}-5\right)^{2}+11\right)$$

To find the opposite of x^{3}-6x^{2}+12x-8, find the opposite of each term.

$$4\left(x^{3}+3x^{2}+3x+1-x^{3}+6x^{2}-12x+8\right)=9\left(\left(2x^{1}-5\right)^{2}+11\right)$$

Combine x^{3} and -x^{3} to get 0.

$$4\left(3x^{2}+3x+1+6x^{2}-12x+8\right)=9\left(\left(2x^{1}-5\right)^{2}+11\right)$$

Combine 3x^{2} and 6x^{2} to get 9x^{2}.

$$4\left(9x^{2}+3x+1-12x+8\right)=9\left(\left(2x^{1}-5\right)^{2}+11\right)$$

Combine 3x and -12x to get -9x.

$$4\left(9x^{2}-9x+1+8\right)=9\left(\left(2x^{1}-5\right)^{2}+11\right)$$

Add 1 and 8 to get 9.

$$4\left(9x^{2}-9x+9\right)=9\left(\left(2x^{1}-5\right)^{2}+11\right)$$

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

$$36x^{2}-36x+36=9\left(\left(2x^{1}-5\right)^{2}+11\right)$$

Calculate x to the power of 1 and get x.

$$36x^{2}-36x+36=9\left(\left(2x-5\right)^{2}+11\right)$$

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

$$36x^{2}-36x+36=9\left(4x^{2}-20x+25+11\right)$$

Add 25 and 11 to get 36.

$$36x^{2}-36x+36=9\left(4x^{2}-20x+36\right)$$

Use the distributive property to multiply 9 by 4x^{2}-20x+36.

$$36x^{2}-36x+36=36x^{2}-180x+324$$

Subtract 36x^{2} from both sides.

$$36x^{2}-36x+36-36x^{2}=-180x+324$$

Combine 36x^{2} and -36x^{2} to get 0.

$$-36x+36=-180x+324$$

Add 180x to both sides.

$$-36x+36+180x=324$$

Combine -36x and 180x to get 144x.

$$144x+36=324$$

Subtract 36 from both sides.

$$144x=324-36$$

Subtract 36 from 324 to get 288.

$$144x=288$$

Divide both sides by 144.

$$x=\frac{288}{144}$$

Divide 288 by 144 to get 2.

$$x=2$$