Solve 2x^3-4x^2-15x+5/x^2-2x-8divx^2+1/9

$$\frac { 2 x ^ { 3 } - 4 x ^ { 2 } - 15 x + 5 } { x ^ { 2 } - 2 x - 8 } \div x ^ { 2 } + \frac { 1 } { 9 }$$

Evaluate

$$\frac{x^{4}+16x^{3}-44x^{2}-135x+45}{9\left(x-4\right)\left(x+2\right)x^{2}}$$

Short Solution Steps

Express \frac{\frac{2x^{3}-4x^{2}-15x+5}{x^{2}-2x-8}}{x^{2}} as a single fraction.

$$\frac{2x^{3}-4x^{2}-15x+5}{\left(x^{2}-2x-8\right)x^{2}}+\frac{1}{9}$$

Factor \left(x^{2}-2x-8\right)x^{2}.

$$\frac{2x^{3}-4x^{2}-15x+5}{\left(x-4\right)\left(x+2\right)x^{2}}+\frac{1}{9}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x+2\right)x^{2} and 9 is 9\left(x-4\right)\left(x+2\right)x^{2}. Multiply \frac{2x^{3}-4x^{2}-15x+5}{\left(x-4\right)\left(x+2\right)x^{2}} times \frac{9}{9}. Multiply \frac{1}{9} times \frac{\left(x-4\right)\left(x+2\right)x^{2}}{\left(x-4\right)\left(x+2\right)x^{2}}.

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

Since \frac{9\left(2x^{3}-4x^{2}-15x+5\right)}{9\left(x-4\right)\left(x+2\right)x^{2}} and \frac{\left(x-4\right)\left(x+2\right)x^{2}}{9\left(x-4\right)\left(x+2\right)x^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 9\left(2x^{3}-4x^{2}-15x+5\right)+\left(x-4\right)\left(x+2\right)x^{2}.

$$\frac{18x^{3}-36x^{2}-135x+45+x^{4}+2x^{3}-4x^{3}-8x^{2}}{9\left(x-4\right)\left(x+2\right)x^{2}}$$

Combine like terms in 18x^{3}-36x^{2}-135x+45+x^{4}+2x^{3}-4x^{3}-8x^{2}.

$$\frac{16x^{3}-44x^{2}-135x+45+x^{4}}{9\left(x-4\right)\left(x+2\right)x^{2}}$$

Expand 9\left(x-4\right)\left(x+2\right)x^{2}.

$$\frac{16x^{3}-44x^{2}-135x+45+x^{4}}{9x^{4}-18x^{3}-72x^{2}}$$

Expand

$$\frac{x^{4}+16x^{3}-44x^{2}-135x+45}{9x^{2}\left(x^{2}-2x-8\right)}$$

Short Solution Steps

Express \frac{\frac{2x^{3}-4x^{2}-15x+5}{x^{2}-2x-8}}{x^{2}} as a single fraction.

$$\frac{2x^{3}-4x^{2}-15x+5}{\left(x^{2}-2x-8\right)x^{2}}+\frac{1}{9}$$

Factor \left(x^{2}-2x-8\right)x^{2}.

$$\frac{2x^{3}-4x^{2}-15x+5}{\left(x-4\right)\left(x+2\right)x^{2}}+\frac{1}{9}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x+2\right)x^{2} and 9 is 9\left(x-4\right)\left(x+2\right)x^{2}. Multiply \frac{2x^{3}-4x^{2}-15x+5}{\left(x-4\right)\left(x+2\right)x^{2}} times \frac{9}{9}. Multiply \frac{1}{9} times \frac{\left(x-4\right)\left(x+2\right)x^{2}}{\left(x-4\right)\left(x+2\right)x^{2}}.

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

Since \frac{9\left(2x^{3}-4x^{2}-15x+5\right)}{9\left(x-4\right)\left(x+2\right)x^{2}} and \frac{\left(x-4\right)\left(x+2\right)x^{2}}{9\left(x-4\right)\left(x+2\right)x^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 9\left(2x^{3}-4x^{2}-15x+5\right)+\left(x-4\right)\left(x+2\right)x^{2}.

$$\frac{18x^{3}-36x^{2}-135x+45+x^{4}+2x^{3}-4x^{3}-8x^{2}}{9\left(x-4\right)\left(x+2\right)x^{2}}$$

Combine like terms in 18x^{3}-36x^{2}-135x+45+x^{4}+2x^{3}-4x^{3}-8x^{2}.

$$\frac{16x^{3}-44x^{2}-135x+45+x^{4}}{9\left(x-4\right)\left(x+2\right)x^{2}}$$

Expand 9\left(x-4\right)\left(x+2\right)x^{2}.

$$\frac{16x^{3}-44x^{2}-135x+45+x^{4}}{9x^{4}-18x^{3}-72x^{2}}$$