Solve {2/3x-[3-4x/5-(4x^2-3/3)+(2x+3)^2/3]+x}=-frac{23}{15}

$$\{ \frac { 2 } { 3 } x - [ \frac { 3 - 4 x } { 5 } - ( \frac { 4 x ^ { 2 } - 3 } { 3 } ) + \frac { ( 2 x + 3 ) ^ { 2 } } { 3 } ] + x \} = - \frac { 23 } { 15 }$$

Solve for x

$$x=-2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 15, the least common multiple of 3,5,15.

$$15\left(\frac{2}{3}x-\left(\frac{3-4x}{5}-\frac{4x^{2}-3}{3}+\frac{\left(2x+3\right)^{2}}{3}\right)\right)+15x=-23$$

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

$$15\left(\frac{2}{3}x-\left(\frac{3-4x}{5}-\frac{4x^{2}-3}{3}+\frac{4x^{2}+12x+9}{3}\right)\right)+15x=-23$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 3 is 15. Multiply \frac{3-4x}{5} times \frac{3}{3}. Multiply \frac{4x^{2}-3}{3} times \frac{5}{5}.

$$15\left(\frac{2}{3}x-\left(\frac{3\left(3-4x\right)}{15}-\frac{5\left(4x^{2}-3\right)}{15}+\frac{4x^{2}+12x+9}{3}\right)\right)+15x=-23$$

Since \frac{3\left(3-4x\right)}{15} and \frac{5\left(4x^{2}-3\right)}{15} have the same denominator, subtract them by subtracting their numerators.

$$15\left(\frac{2}{3}x-\left(\frac{3\left(3-4x\right)-5\left(4x^{2}-3\right)}{15}+\frac{4x^{2}+12x+9}{3}\right)\right)+15x=-23$$

Do the multiplications in 3\left(3-4x\right)-5\left(4x^{2}-3\right).

$$15\left(\frac{2}{3}x-\left(\frac{9-12x-20x^{2}+15}{15}+\frac{4x^{2}+12x+9}{3}\right)\right)+15x=-23$$

Combine like terms in 9-12x-20x^{2}+15.

$$15\left(\frac{2}{3}x-\left(\frac{24-12x-20x^{2}}{15}+\frac{4x^{2}+12x+9}{3}\right)\right)+15x=-23$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 15 and 3 is 15. Multiply \frac{4x^{2}+12x+9}{3} times \frac{5}{5}.

$$15\left(\frac{2}{3}x-\left(\frac{24-12x-20x^{2}}{15}+\frac{5\left(4x^{2}+12x+9\right)}{15}\right)\right)+15x=-23$$

Since \frac{24-12x-20x^{2}}{15} and \frac{5\left(4x^{2}+12x+9\right)}{15} have the same denominator, add them by adding their numerators.

$$15\left(\frac{2}{3}x-\frac{24-12x-20x^{2}+5\left(4x^{2}+12x+9\right)}{15}\right)+15x=-23$$

Do the multiplications in 24-12x-20x^{2}+5\left(4x^{2}+12x+9\right).

$$15\left(\frac{2}{3}x-\frac{24-12x-20x^{2}+20x^{2}+60x+45}{15}\right)+15x=-23$$

Combine like terms in 24-12x-20x^{2}+20x^{2}+60x+45.

$$15\left(\frac{2}{3}x-\frac{69+48x}{15}\right)+15x=-23$$

Use the distributive property to multiply 15 by \frac{2}{3}x-\frac{69+48x}{15}.

$$10x+15\left(-\frac{69+48x}{15}\right)+15x=-23$$

Express 15\left(-\frac{69+48x}{15}\right) as a single fraction.

$$10x+\frac{-15\left(69+48x\right)}{15}+15x=-23$$

Cancel out 15 and 15.

$$10x-\left(69+48x\right)+15x=-23$$

To find the opposite of 69+48x, find the opposite of each term.

$$10x-69-48x+15x=-23$$

Combine 10x and -48x to get -38x.

$$-38x-69+15x=-23$$

Combine -38x and 15x to get -23x.

$$-23x-69=-23$$

Add 69 to both sides.

$$-23x=-23+69$$

Add -23 and 69 to get 46.

$$-23x=46$$

Divide both sides by -23.

$$x=\frac{46}{-23}$$

Divide 46 by -23 to get -2.

$$x=-2$$