Solve (m+3/m+m/-3-m+0/m^2-3m)*m^2-9/m^5-27=

$$( \frac { m + 3 } { m } + \frac { m } { - 3 - m } + \frac { 0 } { m ^ { 2 } - 3 m } ) \cdot \frac { m ^ { 2 } - 9 } { m ^ { 5 } - 27 } =$$

Evaluate

$$\frac{3\left(m-3\right)\left(2m+3\right)}{m\left(m^{5}-27\right)}$$

Short Solution Steps

Zero divided by any non-zero term gives zero.

$$\left(\frac{m+3}{m}+\frac{m}{-3-m}+0\right)\times \frac{m^{2}-9}{m^{5}-27}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m and -3-m is m\left(-m-3\right). Multiply \frac{m+3}{m} times \frac{-m-3}{-m-3}. Multiply \frac{m}{-3-m} times \frac{m}{m}.

$$\left(\frac{\left(m+3\right)\left(-m-3\right)}{m\left(-m-3\right)}+\frac{mm}{m\left(-m-3\right)}+0\right)\times \frac{m^{2}-9}{m^{5}-27}$$

Since \frac{\left(m+3\right)\left(-m-3\right)}{m\left(-m-3\right)} and \frac{mm}{m\left(-m-3\right)} have the same denominator, add them by adding their numerators.

$$\left(\frac{\left(m+3\right)\left(-m-3\right)+mm}{m\left(-m-3\right)}+0\right)\times \frac{m^{2}-9}{m^{5}-27}$$

Do the multiplications in \left(m+3\right)\left(-m-3\right)+mm.

$$\left(\frac{-m^{2}-3m-3m-9+m^{2}}{m\left(-m-3\right)}+0\right)\times \frac{m^{2}-9}{m^{5}-27}$$

Combine like terms in -m^{2}-3m-3m-9+m^{2}.

$$\left(\frac{-6m-9}{m\left(-m-3\right)}+0\right)\times \frac{m^{2}-9}{m^{5}-27}$$

Anything plus zero gives itself.

$$\frac{-6m-9}{m\left(-m-3\right)}\times \frac{m^{2}-9}{m^{5}-27}$$

Multiply \frac{-6m-9}{m\left(-m-3\right)} times \frac{m^{2}-9}{m^{5}-27} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-6m-9\right)\left(m^{2}-9\right)}{m\left(-m-3\right)\left(m^{5}-27\right)}$$

Factor the expressions that are not already factored.

$$\frac{3\left(m-3\right)\left(-2m-3\right)\left(m+3\right)}{m\left(-m-3\right)\left(m^{5}-27\right)}$$

Extract the negative sign in 3+m.

$$\frac{-3\left(m-3\right)\left(-2m-3\right)\left(-m-3\right)}{m\left(-m-3\right)\left(m^{5}-27\right)}$$

Cancel out -m-3 in both numerator and denominator.

$$\frac{-3\left(m-3\right)\left(-2m-3\right)}{m\left(m^{5}-27\right)}$$

Expand the expression.

$$\frac{6m^{2}-9m-27}{m^{6}-27m}$$

Expand

$$\frac{3\left(2m^{2}-3m-9\right)}{m\left(m^{5}-27\right)}$$

Short Solution Steps

Zero divided by any non-zero term gives zero.

$$\left(\frac{m+3}{m}+\frac{m}{-3-m}+0\right)\times \frac{m^{2}-9}{m^{5}-27}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m and -3-m is m\left(-m-3\right). Multiply \frac{m+3}{m} times \frac{-m-3}{-m-3}. Multiply \frac{m}{-3-m} times \frac{m}{m}.

$$\left(\frac{\left(m+3\right)\left(-m-3\right)}{m\left(-m-3\right)}+\frac{mm}{m\left(-m-3\right)}+0\right)\times \frac{m^{2}-9}{m^{5}-27}$$

Since \frac{\left(m+3\right)\left(-m-3\right)}{m\left(-m-3\right)} and \frac{mm}{m\left(-m-3\right)} have the same denominator, add them by adding their numerators.

$$\left(\frac{\left(m+3\right)\left(-m-3\right)+mm}{m\left(-m-3\right)}+0\right)\times \frac{m^{2}-9}{m^{5}-27}$$

Do the multiplications in \left(m+3\right)\left(-m-3\right)+mm.

$$\left(\frac{-m^{2}-3m-3m-9+m^{2}}{m\left(-m-3\right)}+0\right)\times \frac{m^{2}-9}{m^{5}-27}$$

Combine like terms in -m^{2}-3m-3m-9+m^{2}.

$$\left(\frac{-6m-9}{m\left(-m-3\right)}+0\right)\times \frac{m^{2}-9}{m^{5}-27}$$

Anything plus zero gives itself.

$$\frac{-6m-9}{m\left(-m-3\right)}\times \frac{m^{2}-9}{m^{5}-27}$$

Multiply \frac{-6m-9}{m\left(-m-3\right)} times \frac{m^{2}-9}{m^{5}-27} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-6m-9\right)\left(m^{2}-9\right)}{m\left(-m-3\right)\left(m^{5}-27\right)}$$

Factor the expressions that are not already factored.

$$\frac{3\left(m-3\right)\left(-2m-3\right)\left(m+3\right)}{m\left(-m-3\right)\left(m^{5}-27\right)}$$

Extract the negative sign in 3+m.

$$\frac{-3\left(m-3\right)\left(-2m-3\right)\left(-m-3\right)}{m\left(-m-3\right)\left(m^{5}-27\right)}$$

Cancel out -m-3 in both numerator and denominator.

$$\frac{-3\left(m-3\right)\left(-2m-3\right)}{m\left(m^{5}-27\right)}$$

Expand the expression.

$$\frac{6m^{2}-9m-27}{m^{6}-27m}$$