Evaluate
\frac{x^{2}+9x+19}{4\left(x+2\right)\left(x+3\right)}
Expand
\frac{x^{2}+9x+19}{4\left(x+2\right)\left(x+3\right)}
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\frac{\left(2x^{3}-12x^{2}\right)\left(x^{2}+9x+19\right)}{\left(x^{2}-4x-12\right)\left(8x^{3}+24x^{2}\right)}
Divide \frac{2x^{3}-12x^{2}}{x^{2}-4x-12} by \frac{8x^{3}+24x^{2}}{x^{2}+9x+19} by multiplying \frac{2x^{3}-12x^{2}}{x^{2}-4x-12} by the reciprocal of \frac{8x^{3}+24x^{2}}{x^{2}+9x+19}.
\frac{2\left(x-6\right)\left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{9}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{9}{2}\right)\right)x^{2}}{8\left(x-6\right)\left(x+2\right)\left(x+3\right)x^{2}}
Factor the expressions that are not already factored.
\frac{\left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{9}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{9}{2}\right)\right)}{4\left(x+2\right)\left(x+3\right)}
Cancel out 2\left(x-6\right)x^{2} in both numerator and denominator.
\frac{x^{2}+9x+19}{4x^{2}+20x+24}
Expand the expression.
\frac{\left(2x^{3}-12x^{2}\right)\left(x^{2}+9x+19\right)}{\left(x^{2}-4x-12\right)\left(8x^{3}+24x^{2}\right)}
Divide \frac{2x^{3}-12x^{2}}{x^{2}-4x-12} by \frac{8x^{3}+24x^{2}}{x^{2}+9x+19} by multiplying \frac{2x^{3}-12x^{2}}{x^{2}-4x-12} by the reciprocal of \frac{8x^{3}+24x^{2}}{x^{2}+9x+19}.
\frac{2\left(x-6\right)\left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{9}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{9}{2}\right)\right)x^{2}}{8\left(x-6\right)\left(x+2\right)\left(x+3\right)x^{2}}
Factor the expressions that are not already factored.
\frac{\left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{9}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{9}{2}\right)\right)}{4\left(x+2\right)\left(x+3\right)}
Cancel out 2\left(x-6\right)x^{2} in both numerator and denominator.
\frac{x^{2}+9x+19}{4x^{2}+20x+24}
Expand the expression.
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