Solve 3-7x+1/2x-4-x^2-5x-24/2x^2+5x-12

$$3 - \frac { 7 x + 1 } { 2 x - 4 } - \frac { x ^ { 2 } - 5 x - 24 } { 2 x ^ { 2 } + 5 x - 12 }$$

Evaluate

$$\frac{60-25x-17x^{2}-4x^{3}}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}$$

Short Solution Steps

Factor 2x-4.

$$3-\frac{7x+1}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{2x^{2}+5x-12}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{2\left(x-2\right)}{2\left(x-2\right)}.

$$\frac{3\times 2\left(x-2\right)}{2\left(x-2\right)}-\frac{7x+1}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{2x^{2}+5x-12}$$

Since \frac{3\times 2\left(x-2\right)}{2\left(x-2\right)} and \frac{7x+1}{2\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\times 2\left(x-2\right)-\left(7x+1\right)}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{2x^{2}+5x-12}$$

Do the multiplications in 3\times 2\left(x-2\right)-\left(7x+1\right).

$$\frac{6x-12-7x-1}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{2x^{2}+5x-12}$$

Combine like terms in 6x-12-7x-1.

$$\frac{-x-13}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{2x^{2}+5x-12}$$

Factor 2x^{2}+5x-12.

$$\frac{-x-13}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{\left(2x-3\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-2\right) and \left(2x-3\right)\left(x+4\right) is 2\left(x-2\right)\left(2x-3\right)\left(x+4\right). Multiply \frac{-x-13}{2\left(x-2\right)} times \frac{\left(2x-3\right)\left(x+4\right)}{\left(2x-3\right)\left(x+4\right)}. Multiply \frac{x^{2}-5x-24}{\left(2x-3\right)\left(x+4\right)} times \frac{2\left(x-2\right)}{2\left(x-2\right)}.

$$\frac{\left(-x-13\right)\left(2x-3\right)\left(x+4\right)}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}-\frac{\left(x^{2}-5x-24\right)\times 2\left(x-2\right)}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}$$

Since \frac{\left(-x-13\right)\left(2x-3\right)\left(x+4\right)}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)} and \frac{\left(x^{2}-5x-24\right)\times 2\left(x-2\right)}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-x-13\right)\left(2x-3\right)\left(x+4\right)-\left(x^{2}-5x-24\right)\times 2\left(x-2\right)}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}$$

Do the multiplications in \left(-x-13\right)\left(2x-3\right)\left(x+4\right)-\left(x^{2}-5x-24\right)\times 2\left(x-2\right).

$$\frac{-2x^{3}-5x^{2}+12x-26x^{2}-65x+156-2x^{3}+4x^{2}+10x^{2}-20x+48x-96}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}$$

Combine like terms in -2x^{3}-5x^{2}+12x-26x^{2}-65x+156-2x^{3}+4x^{2}+10x^{2}-20x+48x-96.

$$\frac{-4x^{3}-17x^{2}-25x+60}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}$$

Expand 2\left(x-2\right)\left(2x-3\right)\left(x+4\right).

$$\frac{-4x^{3}-17x^{2}-25x+60}{4x^{3}+2x^{2}-44x+48}$$

Expand

$$-\frac{4x^{3}+17x^{2}+25x-60}{2\left(x-2\right)\left(2x^{2}+5x-12\right)}$$

Short Solution Steps

Factor 2x-4.

$$3-\frac{7x+1}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{2x^{2}+5x-12}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{2\left(x-2\right)}{2\left(x-2\right)}.

$$\frac{3\times 2\left(x-2\right)}{2\left(x-2\right)}-\frac{7x+1}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{2x^{2}+5x-12}$$

Since \frac{3\times 2\left(x-2\right)}{2\left(x-2\right)} and \frac{7x+1}{2\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\times 2\left(x-2\right)-\left(7x+1\right)}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{2x^{2}+5x-12}$$

Do the multiplications in 3\times 2\left(x-2\right)-\left(7x+1\right).

$$\frac{6x-12-7x-1}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{2x^{2}+5x-12}$$

Combine like terms in 6x-12-7x-1.

$$\frac{-x-13}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{2x^{2}+5x-12}$$

Factor 2x^{2}+5x-12.

$$\frac{-x-13}{2\left(x-2\right)}-\frac{x^{2}-5x-24}{\left(2x-3\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-2\right) and \left(2x-3\right)\left(x+4\right) is 2\left(x-2\right)\left(2x-3\right)\left(x+4\right). Multiply \frac{-x-13}{2\left(x-2\right)} times \frac{\left(2x-3\right)\left(x+4\right)}{\left(2x-3\right)\left(x+4\right)}. Multiply \frac{x^{2}-5x-24}{\left(2x-3\right)\left(x+4\right)} times \frac{2\left(x-2\right)}{2\left(x-2\right)}.

$$\frac{\left(-x-13\right)\left(2x-3\right)\left(x+4\right)}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}-\frac{\left(x^{2}-5x-24\right)\times 2\left(x-2\right)}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}$$

Since \frac{\left(-x-13\right)\left(2x-3\right)\left(x+4\right)}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)} and \frac{\left(x^{2}-5x-24\right)\times 2\left(x-2\right)}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-x-13\right)\left(2x-3\right)\left(x+4\right)-\left(x^{2}-5x-24\right)\times 2\left(x-2\right)}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}$$

Do the multiplications in \left(-x-13\right)\left(2x-3\right)\left(x+4\right)-\left(x^{2}-5x-24\right)\times 2\left(x-2\right).

$$\frac{-2x^{3}-5x^{2}+12x-26x^{2}-65x+156-2x^{3}+4x^{2}+10x^{2}-20x+48x-96}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}$$

Combine like terms in -2x^{3}-5x^{2}+12x-26x^{2}-65x+156-2x^{3}+4x^{2}+10x^{2}-20x+48x-96.

$$\frac{-4x^{3}-17x^{2}-25x+60}{2\left(x-2\right)\left(2x-3\right)\left(x+4\right)}$$

Expand 2\left(x-2\right)\left(2x-3\right)\left(x+4\right).

$$\frac{-4x^{3}-17x^{2}-25x+60}{4x^{3}+2x^{2}-44x+48}$$