Solve (2m/m+2-m/m-2)divm/m^2-4quadquadquad(5)(a-2/a^2+2a-frac{a-1}{a^2+4a+4})divfrac{a-4}{a+2}

$$( \frac { 2 m } { m + 2 } - \frac { m } { m - 2 } ) \div \frac { m } { m ^ { 2 } - 4 } \quad \quad \quad ( 5 ) ( \frac { a - 2 } { a ^ { 2 } + 2 a } - \frac { a - 1 } { a ^ { 2 } + 4 a + 4 } ) \div \frac { a - 4 } { a + 2 }$$

Evaluate

$$\frac{5\left(m-6\right)}{a\left(a+2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m+2 and m-2 is \left(m-2\right)\left(m+2\right). Multiply \frac{2m}{m+2} times \frac{m-2}{m-2}. Multiply \frac{m}{m-2} times \frac{m+2}{m+2}.

$$\frac{\frac{\frac{2m\left(m-2\right)}{\left(m-2\right)\left(m+2\right)}-\frac{m\left(m+2\right)}{\left(m-2\right)\left(m+2\right)}}{\frac{m}{m^{2}-4}}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Since \frac{2m\left(m-2\right)}{\left(m-2\right)\left(m+2\right)} and \frac{m\left(m+2\right)}{\left(m-2\right)\left(m+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\frac{2m\left(m-2\right)-m\left(m+2\right)}{\left(m-2\right)\left(m+2\right)}}{\frac{m}{m^{2}-4}}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Do the multiplications in 2m\left(m-2\right)-m\left(m+2\right).

$$\frac{\frac{\frac{2m^{2}-4m-m^{2}-2m}{\left(m-2\right)\left(m+2\right)}}{\frac{m}{m^{2}-4}}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Combine like terms in 2m^{2}-4m-m^{2}-2m.

$$\frac{\frac{\frac{m^{2}-6m}{\left(m-2\right)\left(m+2\right)}}{\frac{m}{m^{2}-4}}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Divide \frac{m^{2}-6m}{\left(m-2\right)\left(m+2\right)} by \frac{m}{m^{2}-4} by multiplying \frac{m^{2}-6m}{\left(m-2\right)\left(m+2\right)} by the reciprocal of \frac{m}{m^{2}-4}.

$$\frac{\frac{\left(m^{2}-6m\right)\left(m^{2}-4\right)}{\left(m-2\right)\left(m+2\right)m}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Factor the expressions that are not already factored in \frac{\left(m^{2}-6m\right)\left(m^{2}-4\right)}{\left(m-2\right)\left(m+2\right)m}.

$$\frac{\frac{m\left(m-6\right)\left(m-2\right)\left(m+2\right)}{m\left(m-2\right)\left(m+2\right)}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

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

$$\frac{\left(m-6\right)\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Factor a^{2}+2a. Factor a^{2}+4a+4.

$$\frac{\left(m-6\right)\times 5\left(\frac{a-2}{a\left(a+2\right)}-\frac{a-1}{\left(a+2\right)^{2}}\right)}{\frac{a-4}{a+2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a+2\right) and \left(a+2\right)^{2} is a\left(a+2\right)^{2}. Multiply \frac{a-2}{a\left(a+2\right)} times \frac{a+2}{a+2}. Multiply \frac{a-1}{\left(a+2\right)^{2}} times \frac{a}{a}.

$$\frac{\left(m-6\right)\times 5\left(\frac{\left(a-2\right)\left(a+2\right)}{a\left(a+2\right)^{2}}-\frac{\left(a-1\right)a}{a\left(a+2\right)^{2}}\right)}{\frac{a-4}{a+2}}$$

Since \frac{\left(a-2\right)\left(a+2\right)}{a\left(a+2\right)^{2}} and \frac{\left(a-1\right)a}{a\left(a+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(m-6\right)\times 5\times \frac{\left(a-2\right)\left(a+2\right)-\left(a-1\right)a}{a\left(a+2\right)^{2}}}{\frac{a-4}{a+2}}$$

Do the multiplications in \left(a-2\right)\left(a+2\right)-\left(a-1\right)a.

$$\frac{\left(m-6\right)\times 5\times \frac{a^{2}+2a-2a-4-a^{2}+a}{a\left(a+2\right)^{2}}}{\frac{a-4}{a+2}}$$

Combine like terms in a^{2}+2a-2a-4-a^{2}+a.

$$\frac{\left(m-6\right)\times 5\times \frac{a-4}{a\left(a+2\right)^{2}}}{\frac{a-4}{a+2}}$$

Express \left(m-6\right)\times \frac{a-4}{a\left(a+2\right)^{2}} as a single fraction.

$$\frac{\frac{\left(m-6\right)\left(a-4\right)}{a\left(a+2\right)^{2}}\times 5}{\frac{a-4}{a+2}}$$

Express \frac{\left(m-6\right)\left(a-4\right)}{a\left(a+2\right)^{2}}\times 5 as a single fraction.

$$\frac{\frac{\left(m-6\right)\left(a-4\right)\times 5}{a\left(a+2\right)^{2}}}{\frac{a-4}{a+2}}$$

Divide \frac{\left(m-6\right)\left(a-4\right)\times 5}{a\left(a+2\right)^{2}} by \frac{a-4}{a+2} by multiplying \frac{\left(m-6\right)\left(a-4\right)\times 5}{a\left(a+2\right)^{2}} by the reciprocal of \frac{a-4}{a+2}.

$$\frac{\left(m-6\right)\left(a-4\right)\times 5\left(a+2\right)}{a\left(a+2\right)^{2}\left(a-4\right)}$$

Cancel out \left(a-4\right)\left(a+2\right) in both numerator and denominator.

$$\frac{5\left(m-6\right)}{a\left(a+2\right)}$$

Use the distributive property to multiply 5 by m-6.

$$\frac{5m-30}{a\left(a+2\right)}$$

Use the distributive property to multiply a by a+2.

$$\frac{5m-30}{a^{2}+2a}$$

Expand

$$\frac{5\left(m-6\right)}{a\left(a+2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m+2 and m-2 is \left(m-2\right)\left(m+2\right). Multiply \frac{2m}{m+2} times \frac{m-2}{m-2}. Multiply \frac{m}{m-2} times \frac{m+2}{m+2}.

$$\frac{\frac{\frac{2m\left(m-2\right)}{\left(m-2\right)\left(m+2\right)}-\frac{m\left(m+2\right)}{\left(m-2\right)\left(m+2\right)}}{\frac{m}{m^{2}-4}}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Since \frac{2m\left(m-2\right)}{\left(m-2\right)\left(m+2\right)} and \frac{m\left(m+2\right)}{\left(m-2\right)\left(m+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\frac{2m\left(m-2\right)-m\left(m+2\right)}{\left(m-2\right)\left(m+2\right)}}{\frac{m}{m^{2}-4}}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Do the multiplications in 2m\left(m-2\right)-m\left(m+2\right).

$$\frac{\frac{\frac{2m^{2}-4m-m^{2}-2m}{\left(m-2\right)\left(m+2\right)}}{\frac{m}{m^{2}-4}}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Combine like terms in 2m^{2}-4m-m^{2}-2m.

$$\frac{\frac{\frac{m^{2}-6m}{\left(m-2\right)\left(m+2\right)}}{\frac{m}{m^{2}-4}}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Divide \frac{m^{2}-6m}{\left(m-2\right)\left(m+2\right)} by \frac{m}{m^{2}-4} by multiplying \frac{m^{2}-6m}{\left(m-2\right)\left(m+2\right)} by the reciprocal of \frac{m}{m^{2}-4}.

$$\frac{\frac{\left(m^{2}-6m\right)\left(m^{2}-4\right)}{\left(m-2\right)\left(m+2\right)m}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Factor the expressions that are not already factored in \frac{\left(m^{2}-6m\right)\left(m^{2}-4\right)}{\left(m-2\right)\left(m+2\right)m}.

$$\frac{\frac{m\left(m-6\right)\left(m-2\right)\left(m+2\right)}{m\left(m-2\right)\left(m+2\right)}\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

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

$$\frac{\left(m-6\right)\times 5\left(\frac{a-2}{a^{2}+2a}-\frac{a-1}{a^{2}+4a+4}\right)}{\frac{a-4}{a+2}}$$

Factor a^{2}+2a. Factor a^{2}+4a+4.

$$\frac{\left(m-6\right)\times 5\left(\frac{a-2}{a\left(a+2\right)}-\frac{a-1}{\left(a+2\right)^{2}}\right)}{\frac{a-4}{a+2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a+2\right) and \left(a+2\right)^{2} is a\left(a+2\right)^{2}. Multiply \frac{a-2}{a\left(a+2\right)} times \frac{a+2}{a+2}. Multiply \frac{a-1}{\left(a+2\right)^{2}} times \frac{a}{a}.

$$\frac{\left(m-6\right)\times 5\left(\frac{\left(a-2\right)\left(a+2\right)}{a\left(a+2\right)^{2}}-\frac{\left(a-1\right)a}{a\left(a+2\right)^{2}}\right)}{\frac{a-4}{a+2}}$$

Since \frac{\left(a-2\right)\left(a+2\right)}{a\left(a+2\right)^{2}} and \frac{\left(a-1\right)a}{a\left(a+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(m-6\right)\times 5\times \frac{\left(a-2\right)\left(a+2\right)-\left(a-1\right)a}{a\left(a+2\right)^{2}}}{\frac{a-4}{a+2}}$$

Do the multiplications in \left(a-2\right)\left(a+2\right)-\left(a-1\right)a.

$$\frac{\left(m-6\right)\times 5\times \frac{a^{2}+2a-2a-4-a^{2}+a}{a\left(a+2\right)^{2}}}{\frac{a-4}{a+2}}$$

Combine like terms in a^{2}+2a-2a-4-a^{2}+a.

$$\frac{\left(m-6\right)\times 5\times \frac{a-4}{a\left(a+2\right)^{2}}}{\frac{a-4}{a+2}}$$

Express \left(m-6\right)\times \frac{a-4}{a\left(a+2\right)^{2}} as a single fraction.

$$\frac{\frac{\left(m-6\right)\left(a-4\right)}{a\left(a+2\right)^{2}}\times 5}{\frac{a-4}{a+2}}$$

Express \frac{\left(m-6\right)\left(a-4\right)}{a\left(a+2\right)^{2}}\times 5 as a single fraction.

$$\frac{\frac{\left(m-6\right)\left(a-4\right)\times 5}{a\left(a+2\right)^{2}}}{\frac{a-4}{a+2}}$$

Divide \frac{\left(m-6\right)\left(a-4\right)\times 5}{a\left(a+2\right)^{2}} by \frac{a-4}{a+2} by multiplying \frac{\left(m-6\right)\left(a-4\right)\times 5}{a\left(a+2\right)^{2}} by the reciprocal of \frac{a-4}{a+2}.

$$\frac{\left(m-6\right)\left(a-4\right)\times 5\left(a+2\right)}{a\left(a+2\right)^{2}\left(a-4\right)}$$

Cancel out \left(a-4\right)\left(a+2\right) in both numerator and denominator.

$$\frac{5\left(m-6\right)}{a\left(a+2\right)}$$

Use the distributive property to multiply 5 by m-6.

$$\frac{5m-30}{a\left(a+2\right)}$$

Use the distributive property to multiply a by a+2.

$$\frac{5m-30}{a^{2}+2a}$$