Solve x^2+x-6/2x^2+12divx^3-2x^2/2quad1

$$\frac { x ^ { 2 } + x - 6 } { 2 x ^ { 2 } + 12 } \div \frac { x ^ { 3 } - 2 x ^ { 2 } } { 2 } \quad 1$$

Evaluate

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

Short Solution Steps

Divide \frac{x^{2}+x-6}{2x^{2}+12} by \frac{x^{3}-2x^{2}}{2} by multiplying \frac{x^{2}+x-6}{2x^{2}+12} by the reciprocal of \frac{x^{3}-2x^{2}}{2}.

$$\frac{\left(x^{2}+x-6\right)\times 2}{\left(2x^{2}+12\right)\left(x^{3}-2x^{2}\right)}\times 1$$

Factor the expressions that are not already factored in \frac{\left(x^{2}+x-6\right)\times 2}{\left(2x^{2}+12\right)\left(x^{3}-2x^{2}\right)}.

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

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

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

Express \frac{x+3}{x^{2}\left(x^{2}+6\right)}\times 1 as a single fraction.

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

Use the distributive property to multiply x^{2} by x^{2}+6.

$$\frac{x+3}{x^{4}+6x^{2}}$$

Expand

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

Short Solution Steps

Divide \frac{x^{2}+x-6}{2x^{2}+12} by \frac{x^{3}-2x^{2}}{2} by multiplying \frac{x^{2}+x-6}{2x^{2}+12} by the reciprocal of \frac{x^{3}-2x^{2}}{2}.

$$\frac{\left(x^{2}+x-6\right)\times 2}{\left(2x^{2}+12\right)\left(x^{3}-2x^{2}\right)}\times 1$$

Factor the expressions that are not already factored in \frac{\left(x^{2}+x-6\right)\times 2}{\left(2x^{2}+12\right)\left(x^{3}-2x^{2}\right)}.

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

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

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

Express \frac{x+3}{x^{2}\left(x^{2}+6\right)}\times 1 as a single fraction.

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

Use the distributive property to multiply x^{2} by x^{2}+6.

$$\frac{x+3}{x^{4}+6x^{2}}$$