Solve 3x-2/x^2+14x+48*x^2+12x-12/9x^2-4

$$\frac { 3 x - 2 } { x ^ { 2 } + 14 x + 48 } \times \frac { x ^ { 2 } + 12 x - 12 } { 9 x ^ { 2 } - 4 }$$

Evaluate

$$\frac{x^{2}+12x-12}{\left(3x+2\right)\left(x^{2}+14x+48\right)}$$

Short Solution Steps

Multiply \frac{3x-2}{x^{2}+14x+48} times \frac{x^{2}+12x-12}{9x^{2}-4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(3x-2\right)\left(x^{2}+12x-12\right)}{\left(x^{2}+14x+48\right)\left(9x^{2}-4\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(3x-2\right)\left(x-\left(-4\sqrt{3}-6\right)\right)\left(x-\left(4\sqrt{3}-6\right)\right)}{\left(3x-2\right)\left(x+6\right)\left(x+8\right)\left(3x+2\right)}$$

Cancel out 3x-2 in both numerator and denominator.

$$\frac{\left(x-\left(-4\sqrt{3}-6\right)\right)\left(x-\left(4\sqrt{3}-6\right)\right)}{\left(x+6\right)\left(x+8\right)\left(3x+2\right)}$$

Expand the expression.

$$\frac{x^{2}+12x-12}{3x^{3}+44x^{2}+172x+96}$$

Expand

$$\frac{x^{2}+12x-12}{\left(3x+2\right)\left(x^{2}+14x+48\right)}$$

Short Solution Steps

Multiply \frac{3x-2}{x^{2}+14x+48} times \frac{x^{2}+12x-12}{9x^{2}-4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(3x-2\right)\left(x^{2}+12x-12\right)}{\left(x^{2}+14x+48\right)\left(9x^{2}-4\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(3x-2\right)\left(x-\left(-4\sqrt{3}-6\right)\right)\left(x-\left(4\sqrt{3}-6\right)\right)}{\left(3x-2\right)\left(x+6\right)\left(x+8\right)\left(3x+2\right)}$$

Cancel out 3x-2 in both numerator and denominator.

$$\frac{\left(x-\left(-4\sqrt{3}-6\right)\right)\left(x-\left(4\sqrt{3}-6\right)\right)}{\left(x+6\right)\left(x+8\right)\left(3x+2\right)}$$

Expand the expression.

$$\frac{x^{2}+12x-12}{3x^{3}+44x^{2}+172x+96}$$