Solve 1/3x^2-3{5/x+1-frac{x+4/x^2-3x-4}}

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

Evaluate

$$\frac{4\left(x-6\right)}{3\left(x-4\right)\left(x-1\right)\left(x+1\right)^{2}}$$

Short Solution Steps

Divide 1 by \frac{3x^{2}-3}{\frac{5}{x+1}-\frac{x+4}{x^{2}-3x-4}} by multiplying 1 by the reciprocal of \frac{3x^{2}-3}{\frac{5}{x+1}-\frac{x+4}{x^{2}-3x-4}}.

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

Factor x^{2}-3x-4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and \left(x-4\right)\left(x+1\right) is \left(x-4\right)\left(x+1\right). Multiply \frac{5}{x+1} times \frac{x-4}{x-4}.

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

Since \frac{5\left(x-4\right)}{\left(x-4\right)\left(x+1\right)} and \frac{x+4}{\left(x-4\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{\frac{5x-20-x-4}{\left(x-4\right)\left(x+1\right)}}{3x^{2}-3}$$

Combine like terms in 5x-20-x-4.

$$\frac{\frac{4x-24}{\left(x-4\right)\left(x+1\right)}}{3x^{2}-3}$$

Express \frac{\frac{4x-24}{\left(x-4\right)\left(x+1\right)}}{3x^{2}-3} as a single fraction.

$$\frac{4x-24}{\left(x-4\right)\left(x+1\right)\left(3x^{2}-3\right)}$$

Use the distributive property to multiply x-4 by x+1 and combine like terms.

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

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

$$\frac{4x-24}{3x^{4}-15x^{2}-9x^{3}+9x+12}$$

Expand

$$\frac{4\left(x-6\right)}{3\left(x-4\right)\left(x-1\right)\left(x+1\right)^{2}}$$

Short Solution Steps

Divide 1 by \frac{3x^{2}-3}{\frac{5}{x+1}-\frac{x+4}{x^{2}-3x-4}} by multiplying 1 by the reciprocal of \frac{3x^{2}-3}{\frac{5}{x+1}-\frac{x+4}{x^{2}-3x-4}}.

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

Factor x^{2}-3x-4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and \left(x-4\right)\left(x+1\right) is \left(x-4\right)\left(x+1\right). Multiply \frac{5}{x+1} times \frac{x-4}{x-4}.

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

Since \frac{5\left(x-4\right)}{\left(x-4\right)\left(x+1\right)} and \frac{x+4}{\left(x-4\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{\frac{5x-20-x-4}{\left(x-4\right)\left(x+1\right)}}{3x^{2}-3}$$

Combine like terms in 5x-20-x-4.

$$\frac{\frac{4x-24}{\left(x-4\right)\left(x+1\right)}}{3x^{2}-3}$$

Express \frac{\frac{4x-24}{\left(x-4\right)\left(x+1\right)}}{3x^{2}-3} as a single fraction.

$$\frac{4x-24}{\left(x-4\right)\left(x+1\right)\left(3x^{2}-3\right)}$$

Use the distributive property to multiply x-4 by x+1 and combine like terms.

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

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

$$\frac{4x-24}{3x^{4}-15x^{2}-9x^{3}+9x+12}$$