Simplify \( \frac{2}{z+2}+\frac{4}{5 z}+\frac{4}{z^{2}+2 z} \)

Calculate or simplify the expression \( (2/(z+2))+4/(5*z)+4/(z^2+2*z) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{2}{\left(z+2\right)}\right)+\frac{4}{5z}+\frac{4}{\left(z^{2}+2z\right)}\) - step1: Remove the parentheses: \(\left(\frac{2}{z+2}\right)+\frac{4}{5z}+\frac{4}{z^{2}+2z}\) - step2: Remove the parentheses: \(\frac{2}{z+2}+\frac{4}{5z}+\frac{4}{z^{2}+2z}\) - step3: Factor the expression: \(\frac{2}{z+2}+\frac{4}{5z}+\frac{4}{z\left(z+2\right)}\) - step4: Reduce fractions to a common denominator: \(\frac{2\times 5z}{\left(z+2\right)\times 5z}+\frac{4\left(z+2\right)}{5z\left(z+2\right)}+\frac{4\times 5}{z\left(z+2\right)\times 5}\) - step5: Reorder the terms: \(\frac{2\times 5z}{5\left(z+2\right)z}+\frac{4\left(z+2\right)}{5z\left(z+2\right)}+\frac{4\times 5}{z\left(z+2\right)\times 5}\) - step6: Reorder the terms: \(\frac{2\times 5z}{5\left(z+2\right)z}+\frac{4\left(z+2\right)}{5z\left(z+2\right)}+\frac{4\times 5}{5z\left(z+2\right)}\) - step7: Rewrite the expression: \(\frac{2\times 5z}{5\left(z+2\right)z}+\frac{4\left(z+2\right)}{5\left(z+2\right)z}+\frac{4\times 5}{5\left(z+2\right)z}\) - step8: Transform the expression: \(\frac{2\times 5z+4\left(z+2\right)+4\times 5}{5\left(z+2\right)z}\) - step9: Multiply the terms: \(\frac{10z+4\left(z+2\right)+4\times 5}{5\left(z+2\right)z}\) - step10: Multiply the terms: \(\frac{10z+4z+8+4\times 5}{5\left(z+2\right)z}\) - step11: Multiply the numbers: \(\frac{10z+4z+8+20}{5\left(z+2\right)z}\) - step12: Add the terms: \(\frac{14z+28}{5\left(z+2\right)z}\) - step13: Factor the expression: \(\frac{14\left(z+2\right)}{5\left(z+2\right)z}\) - step14: Reduce the fraction: \(\frac{14}{5z}\) The simplified expression is \( \frac{14}{5z} \).