Solve [(3ab/ab+2-3ab/ab-2)frac{a^{2}b^{2}-4}{9ab}+a^2+5a+6/a^2+2a-3]:10^2-6a+3/3a^2-6a+3

$$[ ( \frac { 3 a b } { a b + 2 } - \frac { 3 a b } { a b - 2 } ) \frac { a ^ { 2 } b ^ { 2 } - 4 } { 9 a b } + \frac { a ^ { 2 } + 5 a + 6 } { a ^ { 2 } + 2 a - 3 } ] : \frac { 10 ^ { 2 } - 6 a + 3 } { 3 a ^ { 2 } - 6 a + 3 }$$

Evaluate

$$\frac{\left(10-a\right)\left(a-1\right)}{103-6a}$$

Short Solution Steps

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

$$\frac{\left(\frac{3ab\left(ab-2\right)}{\left(ab-2\right)\left(ab+2\right)}-\frac{3ab\left(ab+2\right)}{\left(ab-2\right)\left(ab+2\right)}\right)\times \frac{a^{2}b^{2}-4}{9ab}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Since \frac{3ab\left(ab-2\right)}{\left(ab-2\right)\left(ab+2\right)} and \frac{3ab\left(ab+2\right)}{\left(ab-2\right)\left(ab+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3ab\left(ab-2\right)-3ab\left(ab+2\right)}{\left(ab-2\right)\left(ab+2\right)}\times \frac{a^{2}b^{2}-4}{9ab}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Do the multiplications in 3ab\left(ab-2\right)-3ab\left(ab+2\right).

$$\frac{\frac{3a^{2}b^{2}-6ab-3a^{2}b^{2}-6ab}{\left(ab-2\right)\left(ab+2\right)}\times \frac{a^{2}b^{2}-4}{9ab}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Combine like terms in 3a^{2}b^{2}-6ab-3a^{2}b^{2}-6ab.

$$\frac{\frac{-12ab}{\left(ab-2\right)\left(ab+2\right)}\times \frac{a^{2}b^{2}-4}{9ab}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Multiply \frac{-12ab}{\left(ab-2\right)\left(ab+2\right)} times \frac{a^{2}b^{2}-4}{9ab} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{-12ab\left(a^{2}b^{2}-4\right)}{\left(ab-2\right)\left(ab+2\right)\times 9ab}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Cancel out 3ab in both numerator and denominator.

$$\frac{\frac{-4\left(a^{2}b^{2}-4\right)}{3\left(ab-2\right)\left(ab+2\right)}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Factor the expressions that are not already factored in \frac{a^{2}+5a+6}{a^{2}+2a-3}.

$$\frac{\frac{-4\left(a^{2}b^{2}-4\right)}{3\left(ab-2\right)\left(ab+2\right)}+\frac{\left(a+2\right)\left(a+3\right)}{\left(a-1\right)\left(a+3\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Cancel out a+3 in both numerator and denominator.

$$\frac{\frac{-4\left(a^{2}b^{2}-4\right)}{3\left(ab-2\right)\left(ab+2\right)}+\frac{a+2}{a-1}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

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

$$\frac{\frac{-4\left(a^{2}b^{2}-4\right)\left(a-1\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}+\frac{\left(a+2\right)\times 3\left(ab-2\right)\left(ab+2\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Since \frac{-4\left(a^{2}b^{2}-4\right)\left(a-1\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)} and \frac{\left(a+2\right)\times 3\left(ab-2\right)\left(ab+2\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-4\left(a^{2}b^{2}-4\right)\left(a-1\right)+\left(a+2\right)\times 3\left(ab-2\right)\left(ab+2\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Do the multiplications in -4\left(a^{2}b^{2}-4\right)\left(a-1\right)+\left(a+2\right)\times 3\left(ab-2\right)\left(ab+2\right).

$$\frac{\frac{-4a^{3}b^{2}+4a^{2}b^{2}+16a-16-12a+3a^{3}b^{2}+6a^{2}b^{2}-24}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Combine like terms in -4a^{3}b^{2}+4a^{2}b^{2}+16a-16-12a+3a^{3}b^{2}+6a^{2}b^{2}-24.

$$\frac{\frac{4a-a^{3}b^{2}+10a^{2}b^{2}-40}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Factor the expressions that are not already factored in \frac{4a-a^{3}b^{2}+10a^{2}b^{2}-40}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}.

$$\frac{\frac{\left(-a+10\right)\left(ab-2\right)\left(ab+2\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

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

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

Calculate 10 to the power of 2 and get 100.

$$\frac{\frac{-a+10}{3\left(a-1\right)}}{\frac{100-6a+3}{3a^{2}-6a+3}}$$

Add 100 and 3 to get 103.

$$\frac{\frac{-a+10}{3\left(a-1\right)}}{\frac{103-6a}{3a^{2}-6a+3}}$$

Divide \frac{-a+10}{3\left(a-1\right)} by \frac{103-6a}{3a^{2}-6a+3} by multiplying \frac{-a+10}{3\left(a-1\right)} by the reciprocal of \frac{103-6a}{3a^{2}-6a+3}.

$$\frac{\left(-a+10\right)\left(3a^{2}-6a+3\right)}{3\left(a-1\right)\left(103-6a\right)}$$

Factor the expressions that are not already factored.

$$\frac{3\left(-a+10\right)\left(a-1\right)^{2}}{3\left(a-1\right)\left(-6a+103\right)}$$

Cancel out 3\left(a-1\right) in both numerator and denominator.

$$\frac{\left(a-1\right)\left(-a+10\right)}{-6a+103}$$

Expand the expression.

$$\frac{-a^{2}+11a-10}{-6a+103}$$

Expand

$$-\frac{a^{2}-11a+10}{103-6a}$$

Short Solution Steps

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

$$\frac{\left(\frac{3ab\left(ab-2\right)}{\left(ab-2\right)\left(ab+2\right)}-\frac{3ab\left(ab+2\right)}{\left(ab-2\right)\left(ab+2\right)}\right)\times \frac{a^{2}b^{2}-4}{9ab}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Since \frac{3ab\left(ab-2\right)}{\left(ab-2\right)\left(ab+2\right)} and \frac{3ab\left(ab+2\right)}{\left(ab-2\right)\left(ab+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3ab\left(ab-2\right)-3ab\left(ab+2\right)}{\left(ab-2\right)\left(ab+2\right)}\times \frac{a^{2}b^{2}-4}{9ab}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Do the multiplications in 3ab\left(ab-2\right)-3ab\left(ab+2\right).

$$\frac{\frac{3a^{2}b^{2}-6ab-3a^{2}b^{2}-6ab}{\left(ab-2\right)\left(ab+2\right)}\times \frac{a^{2}b^{2}-4}{9ab}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Combine like terms in 3a^{2}b^{2}-6ab-3a^{2}b^{2}-6ab.

$$\frac{\frac{-12ab}{\left(ab-2\right)\left(ab+2\right)}\times \frac{a^{2}b^{2}-4}{9ab}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Multiply \frac{-12ab}{\left(ab-2\right)\left(ab+2\right)} times \frac{a^{2}b^{2}-4}{9ab} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{-12ab\left(a^{2}b^{2}-4\right)}{\left(ab-2\right)\left(ab+2\right)\times 9ab}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Cancel out 3ab in both numerator and denominator.

$$\frac{\frac{-4\left(a^{2}b^{2}-4\right)}{3\left(ab-2\right)\left(ab+2\right)}+\frac{a^{2}+5a+6}{a^{2}+2a-3}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Factor the expressions that are not already factored in \frac{a^{2}+5a+6}{a^{2}+2a-3}.

$$\frac{\frac{-4\left(a^{2}b^{2}-4\right)}{3\left(ab-2\right)\left(ab+2\right)}+\frac{\left(a+2\right)\left(a+3\right)}{\left(a-1\right)\left(a+3\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Cancel out a+3 in both numerator and denominator.

$$\frac{\frac{-4\left(a^{2}b^{2}-4\right)}{3\left(ab-2\right)\left(ab+2\right)}+\frac{a+2}{a-1}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

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

$$\frac{\frac{-4\left(a^{2}b^{2}-4\right)\left(a-1\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}+\frac{\left(a+2\right)\times 3\left(ab-2\right)\left(ab+2\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Since \frac{-4\left(a^{2}b^{2}-4\right)\left(a-1\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)} and \frac{\left(a+2\right)\times 3\left(ab-2\right)\left(ab+2\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-4\left(a^{2}b^{2}-4\right)\left(a-1\right)+\left(a+2\right)\times 3\left(ab-2\right)\left(ab+2\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Do the multiplications in -4\left(a^{2}b^{2}-4\right)\left(a-1\right)+\left(a+2\right)\times 3\left(ab-2\right)\left(ab+2\right).

$$\frac{\frac{-4a^{3}b^{2}+4a^{2}b^{2}+16a-16-12a+3a^{3}b^{2}+6a^{2}b^{2}-24}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Combine like terms in -4a^{3}b^{2}+4a^{2}b^{2}+16a-16-12a+3a^{3}b^{2}+6a^{2}b^{2}-24.

$$\frac{\frac{4a-a^{3}b^{2}+10a^{2}b^{2}-40}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

Factor the expressions that are not already factored in \frac{4a-a^{3}b^{2}+10a^{2}b^{2}-40}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}.

$$\frac{\frac{\left(-a+10\right)\left(ab-2\right)\left(ab+2\right)}{3\left(a-1\right)\left(ab-2\right)\left(ab+2\right)}}{\frac{10^{2}-6a+3}{3a^{2}-6a+3}}$$

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

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

Calculate 10 to the power of 2 and get 100.

$$\frac{\frac{-a+10}{3\left(a-1\right)}}{\frac{100-6a+3}{3a^{2}-6a+3}}$$

Add 100 and 3 to get 103.

$$\frac{\frac{-a+10}{3\left(a-1\right)}}{\frac{103-6a}{3a^{2}-6a+3}}$$

Divide \frac{-a+10}{3\left(a-1\right)} by \frac{103-6a}{3a^{2}-6a+3} by multiplying \frac{-a+10}{3\left(a-1\right)} by the reciprocal of \frac{103-6a}{3a^{2}-6a+3}.

$$\frac{\left(-a+10\right)\left(3a^{2}-6a+3\right)}{3\left(a-1\right)\left(103-6a\right)}$$

Factor the expressions that are not already factored.

$$\frac{3\left(-a+10\right)\left(a-1\right)^{2}}{3\left(a-1\right)\left(-6a+103\right)}$$

Cancel out 3\left(a-1\right) in both numerator and denominator.

$$\frac{\left(a-1\right)\left(-a+10\right)}{-6a+103}$$

Expand the expression.

$$\frac{-a^{2}+11a-10}{-6a+103}$$