Solve 1/12x*left(4x+4x^2right)^3-left(4x+4x^2right)^2+3x*(4x+4{x}^2)=y

$$\frac{ 1 }{ 12x } \times { \left(4x+4 { x }^{ 2 } \right) }^{ 3 } - { \left(4x+4 { x }^{ 2 } \right) }^{ 2 } +3x \times (4x+4 { x }^{ 2 } )=y$$

Solve for y

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

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 12x.

$$12x\left(\frac{1}{12x}\left(4x+4x^{2}\right)^{3}-\left(4x+4x^{2}\right)^{2}\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

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

$$12x\left(\frac{1}{12x}\left(64x^{3}+192x^{2}x^{2}+192x\left(x^{2}\right)^{2}+64\left(x^{2}\right)^{3}\right)-\left(4x+4x^{2}\right)^{2}\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$12x\left(\frac{1}{12x}\left(64x^{3}+192x^{4}+192x\left(x^{2}\right)^{2}+64\left(x^{2}\right)^{3}\right)-\left(4x+4x^{2}\right)^{2}\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$12x\left(\frac{1}{12x}\left(64x^{3}+192x^{4}+192xx^{4}+64\left(x^{2}\right)^{3}\right)-\left(4x+4x^{2}\right)^{2}\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$12x\left(\frac{1}{12x}\left(64x^{3}+192x^{4}+192x^{5}+64\left(x^{2}\right)^{3}\right)-\left(4x+4x^{2}\right)^{2}\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

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

$$12x\left(\frac{1}{12x}\left(64x^{3}+192x^{4}+192x^{5}+64x^{6}\right)-\left(4x+4x^{2}\right)^{2}\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

Express \frac{1}{12x}\left(64x^{3}+192x^{4}+192x^{5}+64x^{6}\right) as a single fraction.

$$12x\left(\frac{64x^{3}+192x^{4}+192x^{5}+64x^{6}}{12x}-\left(4x+4x^{2}\right)^{2}\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

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

$$12x\left(\frac{64x^{3}+192x^{4}+192x^{5}+64x^{6}}{12x}-\left(16x^{2}+32xx^{2}+16\left(x^{2}\right)^{2}\right)\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$12x\left(\frac{64x^{3}+192x^{4}+192x^{5}+64x^{6}}{12x}-\left(16x^{2}+32x^{3}+16\left(x^{2}\right)^{2}\right)\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$12x\left(\frac{64x^{3}+192x^{4}+192x^{5}+64x^{6}}{12x}-\left(16x^{2}+32x^{3}+16x^{4}\right)\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

To find the opposite of 16x^{2}+32x^{3}+16x^{4}, find the opposite of each term.

$$12x\left(\frac{64x^{3}+192x^{4}+192x^{5}+64x^{6}}{12x}-16x^{2}-32x^{3}-16x^{4}\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -16x^{2}-32x^{3}-16x^{4} times \frac{12x}{12x}.

$$12x\left(\frac{64x^{3}+192x^{4}+192x^{5}+64x^{6}}{12x}+\frac{\left(-16x^{2}-32x^{3}-16x^{4}\right)\times 12x}{12x}\right)+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

Since \frac{64x^{3}+192x^{4}+192x^{5}+64x^{6}}{12x} and \frac{\left(-16x^{2}-32x^{3}-16x^{4}\right)\times 12x}{12x} have the same denominator, add them by adding their numerators.

$$12x\times \frac{64x^{3}+192x^{4}+192x^{5}+64x^{6}+\left(-16x^{2}-32x^{3}-16x^{4}\right)\times 12x}{12x}+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

Do the multiplications in 64x^{3}+192x^{4}+192x^{5}+64x^{6}+\left(-16x^{2}-32x^{3}-16x^{4}\right)\times 12x.

$$12x\times \frac{64x^{3}+192x^{4}+192x^{5}+64x^{6}-192x^{3}-384x^{4}-192x^{5}}{12x}+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

Combine like terms in 64x^{3}+192x^{4}+192x^{5}+64x^{6}-192x^{3}-384x^{4}-192x^{5}.

$$12x\times \frac{-128x^{3}-192x^{4}+64x^{6}}{12x}+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

Factor the expressions that are not already factored in \frac{-128x^{3}-192x^{4}+64x^{6}}{12x}.

$$12x\times \frac{64\left(x-2\right)\left(x+1\right)^{2}x^{3}}{12x}+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

Cancel out 4x in both numerator and denominator.

$$12x\times \frac{16\left(x-2\right)x^{2}\left(x+1\right)^{2}}{3}+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

Cancel out 3, the greatest common factor in 12 and 3.

$$4\times 16\left(x-2\right)x^{2}\left(x+1\right)^{2}x+3x\left(4x+4x^{2}\right)\times 12x=y\times 12x$$

Multiply x and x to get x^{2}.

$$4\times 16\left(x-2\right)x^{2}\left(x+1\right)^{2}x+3x^{2}\left(4x+4x^{2}\right)\times 12=y\times 12x$$

Multiply 3 and 12 to get 36.

$$4\times 16\left(x-2\right)x^{2}\left(x+1\right)^{2}x+36x^{2}\left(4x+4x^{2}\right)=y\times 12x$$

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

$$4\times 16\left(x-2\right)x^{2}\left(x+1\right)^{2}x+144x^{3}+144x^{4}=y\times 12x$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$4\times 16\left(x-2\right)x^{3}\left(x+1\right)^{2}+144x^{3}+144x^{4}=y\times 12x$$

Multiply 4 and 16 to get 64.

$$64\left(x-2\right)x^{3}\left(x+1\right)^{2}+144x^{3}+144x^{4}=y\times 12x$$

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

$$64\left(x-2\right)x^{3}\left(x^{2}+2x+1\right)+144x^{3}+144x^{4}=y\times 12x$$

Use the distributive property to multiply 64 by x-2.

$$\left(64x-128\right)x^{3}\left(x^{2}+2x+1\right)+144x^{3}+144x^{4}=y\times 12x$$

Use the distributive property to multiply 64x-128 by x^{3}.

$$\left(64x^{4}-128x^{3}\right)\left(x^{2}+2x+1\right)+144x^{3}+144x^{4}=y\times 12x$$

Use the distributive property to multiply 64x^{4}-128x^{3} by x^{2}+2x+1 and combine like terms.

$$64x^{6}-192x^{4}-128x^{3}+144x^{3}+144x^{4}=y\times 12x$$

Combine -128x^{3} and 144x^{3} to get 16x^{3}.

$$64x^{6}-192x^{4}+16x^{3}+144x^{4}=y\times 12x$$

Combine -192x^{4} and 144x^{4} to get -48x^{4}.

$$64x^{6}-48x^{4}+16x^{3}=y\times 12x$$

Swap sides so that all variable terms are on the left hand side.

$$y\times 12x=64x^{6}-48x^{4}+16x^{3}$$

The equation is in standard form.

$$12xy=64x^{6}-48x^{4}+16x^{3}$$

Divide both sides by 12x.

$$\frac{12xy}{12x}=\frac{16\left(x+1\right)\left(2x-1\right)^{2}x^{3}}{12x}$$

Dividing by 12x undoes the multiplication by 12x.

$$y=\frac{16\left(x+1\right)\left(2x-1\right)^{2}x^{3}}{12x}$$

Divide 16\left(1+x\right)x^{3}\left(-1+2x\right)^{2} by 12x.

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