Solve 4x-3/3-2x-4+5x/3+2x-3+x-10x^2/4x^2-9

$$\frac { 4 x - 3 } { 3 - 2 x } - \frac { 4 + 5 x } { 3 + 2 x } - \frac { 3 + x - 10 x ^ { 2 } } { 4 x ^ { 2 } - 9 }$$

Evaluate

$$-2$$

Short Solution Steps

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

$$\frac{\left(4x-3\right)\left(2x+3\right)}{\left(-2x+3\right)\left(2x+3\right)}-\frac{\left(4+5x\right)\left(-2x+3\right)}{\left(-2x+3\right)\left(2x+3\right)}-\frac{3+x-10x^{2}}{4x^{2}-9}$$

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

$$\frac{\left(4x-3\right)\left(2x+3\right)-\left(4+5x\right)\left(-2x+3\right)}{\left(-2x+3\right)\left(2x+3\right)}-\frac{3+x-10x^{2}}{4x^{2}-9}$$

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

$$\frac{8x^{2}+12x-6x-9+8x-12+10x^{2}-15x}{\left(-2x+3\right)\left(2x+3\right)}-\frac{3+x-10x^{2}}{4x^{2}-9}$$

Combine like terms in 8x^{2}+12x-6x-9+8x-12+10x^{2}-15x.

$$\frac{18x^{2}-x-21}{\left(-2x+3\right)\left(2x+3\right)}-\frac{3+x-10x^{2}}{4x^{2}-9}$$

Factor 4x^{2}-9.

$$\frac{18x^{2}-x-21}{\left(-2x+3\right)\left(2x+3\right)}-\frac{3+x-10x^{2}}{\left(2x-3\right)\left(2x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(-2x+3\right)\left(2x+3\right) and \left(2x-3\right)\left(2x+3\right) is \left(2x-3\right)\left(2x+3\right). Multiply \frac{18x^{2}-x-21}{\left(-2x+3\right)\left(2x+3\right)} times \frac{-1}{-1}.

$$\frac{-\left(18x^{2}-x-21\right)}{\left(2x-3\right)\left(2x+3\right)}-\frac{3+x-10x^{2}}{\left(2x-3\right)\left(2x+3\right)}$$

Since \frac{-\left(18x^{2}-x-21\right)}{\left(2x-3\right)\left(2x+3\right)} and \frac{3+x-10x^{2}}{\left(2x-3\right)\left(2x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(18x^{2}-x-21\right)-\left(3+x-10x^{2}\right)}{\left(2x-3\right)\left(2x+3\right)}$$

Do the multiplications in -\left(18x^{2}-x-21\right)-\left(3+x-10x^{2}\right).

$$\frac{-18x^{2}+x+21-3-x+10x^{2}}{\left(2x-3\right)\left(2x+3\right)}$$

Combine like terms in -18x^{2}+x+21-3-x+10x^{2}.

$$\frac{-8x^{2}+18}{\left(2x-3\right)\left(2x+3\right)}$$

Factor the expressions that are not already factored in \frac{-8x^{2}+18}{\left(2x-3\right)\left(2x+3\right)}.

$$\frac{2\left(-2x-3\right)\left(2x-3\right)}{\left(2x-3\right)\left(2x+3\right)}$$

Extract the negative sign in -3-2x.

$$\frac{-2\left(2x-3\right)\left(2x+3\right)}{\left(2x-3\right)\left(2x+3\right)}$$

Cancel out \left(2x-3\right)\left(2x+3\right) in both numerator and denominator.

$$-2$$

Factor

$$-2$$