Solve 3m-2/10m^2-1-6m+5/8m^2-6m+1

$$\frac { 3 m - 2 } { 10 m ^ { 2 } - 1 } - \frac { 6 m + 5 } { 8 m ^ { 2 } - 6 m + 1 }$$

Evaluate

$$\frac{3\left(1+7m-28m^{2}-12m^{3}\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}$$

Short Solution Steps

Factor 8m^{2}-6m+1.

$$\frac{3m-2}{10m^{2}-1}-\frac{6m+5}{\left(2m-1\right)\left(4m-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 10m^{2}-1 and \left(2m-1\right)\left(4m-1\right) is \left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right). Multiply \frac{3m-2}{10m^{2}-1} times \frac{\left(2m-1\right)\left(4m-1\right)}{\left(2m-1\right)\left(4m-1\right)}. Multiply \frac{6m+5}{\left(2m-1\right)\left(4m-1\right)} times \frac{10m^{2}-1}{10m^{2}-1}.

$$\frac{\left(3m-2\right)\left(2m-1\right)\left(4m-1\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}-\frac{\left(6m+5\right)\left(10m^{2}-1\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}$$

Since \frac{\left(3m-2\right)\left(2m-1\right)\left(4m-1\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)} and \frac{\left(6m+5\right)\left(10m^{2}-1\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3m-2\right)\left(2m-1\right)\left(4m-1\right)-\left(6m+5\right)\left(10m^{2}-1\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}$$

Do the multiplications in \left(3m-2\right)\left(2m-1\right)\left(4m-1\right)-\left(6m+5\right)\left(10m^{2}-1\right).

$$\frac{24m^{3}-18m^{2}+3m-16m^{2}+12m-2-60m^{3}+6m-50m^{2}+5}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}$$

Combine like terms in 24m^{3}-18m^{2}+3m-16m^{2}+12m-2-60m^{3}+6m-50m^{2}+5.

$$\frac{-36m^{3}-84m^{2}+21m+3}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}$$

Expand \left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right).

$$\frac{-36m^{3}-84m^{2}+21m+3}{80m^{4}-60m^{3}+2m^{2}+6m-1}$$

Expand

$$-\frac{3\left(12m^{3}+28m^{2}-7m-1\right)}{\left(10m^{2}-1\right)\left(8m^{2}-6m+1\right)}$$

Short Solution Steps

Factor 8m^{2}-6m+1.

$$\frac{3m-2}{10m^{2}-1}-\frac{6m+5}{\left(2m-1\right)\left(4m-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 10m^{2}-1 and \left(2m-1\right)\left(4m-1\right) is \left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right). Multiply \frac{3m-2}{10m^{2}-1} times \frac{\left(2m-1\right)\left(4m-1\right)}{\left(2m-1\right)\left(4m-1\right)}. Multiply \frac{6m+5}{\left(2m-1\right)\left(4m-1\right)} times \frac{10m^{2}-1}{10m^{2}-1}.

$$\frac{\left(3m-2\right)\left(2m-1\right)\left(4m-1\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}-\frac{\left(6m+5\right)\left(10m^{2}-1\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}$$

Since \frac{\left(3m-2\right)\left(2m-1\right)\left(4m-1\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)} and \frac{\left(6m+5\right)\left(10m^{2}-1\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3m-2\right)\left(2m-1\right)\left(4m-1\right)-\left(6m+5\right)\left(10m^{2}-1\right)}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}$$

Do the multiplications in \left(3m-2\right)\left(2m-1\right)\left(4m-1\right)-\left(6m+5\right)\left(10m^{2}-1\right).

$$\frac{24m^{3}-18m^{2}+3m-16m^{2}+12m-2-60m^{3}+6m-50m^{2}+5}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}$$

Combine like terms in 24m^{3}-18m^{2}+3m-16m^{2}+12m-2-60m^{3}+6m-50m^{2}+5.

$$\frac{-36m^{3}-84m^{2}+21m+3}{\left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right)}$$

Expand \left(2m-1\right)\left(4m-1\right)\left(10m^{2}-1\right).

$$\frac{-36m^{3}-84m^{2}+21m+3}{80m^{4}-60m^{3}+2m^{2}+6m-1}$$