Solve 4a+6b/a+b+6a+4b/a-b+4a^2+6b^2/b^2-a^2+frac{4{b}^{2}-6{a}^{2}}{{a}^{2}+{b}^2}-frac{20{b}^4}{{b}^4-{a}^4}

$$\frac{ 4a+6b }{ a+b } + \frac{ 6a+4b }{ a-b } + \frac{ 4 { a }^{ 2 } +6 { b }^{ 2 } }{ { b }^{ 2 } - { a }^{ 2 } } + \frac{ 4 { b }^{ 2 } -6 { a }^{ 2 } }{ { a }^{ 2 } + { b }^{ 2 } } - \frac{ 20 { b }^{ 4 } }{ { b }^{ 4 } - { a }^{ 4 } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+b and a-b is \left(a+b\right)\left(a-b\right). Multiply \frac{4a+6b}{a+b} times \frac{a-b}{a-b}. Multiply \frac{6a+4b}{a-b} times \frac{a+b}{a+b}.

$$\frac{\left(4a+6b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}+\frac{\left(6a+4b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}+\frac{4a^{2}+6b^{2}}{b^{2}-a^{2}}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Since \frac{\left(4a+6b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)} and \frac{\left(6a+4b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4a+6b\right)\left(a-b\right)+\left(6a+4b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}+\frac{4a^{2}+6b^{2}}{b^{2}-a^{2}}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Do the multiplications in \left(4a+6b\right)\left(a-b\right)+\left(6a+4b\right)\left(a+b\right).

$$\frac{4a^{2}-4ab+6ba-6b^{2}+6a^{2}+6ba+4ba+4b^{2}}{\left(a+b\right)\left(a-b\right)}+\frac{4a^{2}+6b^{2}}{b^{2}-a^{2}}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Combine like terms in 4a^{2}-4ab+6ba-6b^{2}+6a^{2}+6ba+4ba+4b^{2}.

$$\frac{10a^{2}-2b^{2}+12ab}{\left(a+b\right)\left(a-b\right)}+\frac{4a^{2}+6b^{2}}{b^{2}-a^{2}}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Factor b^{2}-a^{2}.

$$\frac{10a^{2}-2b^{2}+12ab}{\left(a+b\right)\left(a-b\right)}+\frac{4a^{2}+6b^{2}}{\left(a+b\right)\left(-a+b\right)}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+b\right)\left(a-b\right) and \left(a+b\right)\left(-a+b\right) is \left(a+b\right)\left(-a+b\right). Multiply \frac{10a^{2}-2b^{2}+12ab}{\left(a+b\right)\left(a-b\right)} times \frac{-1}{-1}.

$$\frac{-\left(10a^{2}-2b^{2}+12ab\right)}{\left(a+b\right)\left(-a+b\right)}+\frac{4a^{2}+6b^{2}}{\left(a+b\right)\left(-a+b\right)}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Since \frac{-\left(10a^{2}-2b^{2}+12ab\right)}{\left(a+b\right)\left(-a+b\right)} and \frac{4a^{2}+6b^{2}}{\left(a+b\right)\left(-a+b\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(10a^{2}-2b^{2}+12ab\right)+4a^{2}+6b^{2}}{\left(a+b\right)\left(-a+b\right)}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Do the multiplications in -\left(10a^{2}-2b^{2}+12ab\right)+4a^{2}+6b^{2}.

$$\frac{-10a^{2}+2b^{2}-12ab+4a^{2}+6b^{2}}{\left(a+b\right)\left(-a+b\right)}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Combine like terms in -10a^{2}+2b^{2}-12ab+4a^{2}+6b^{2}.

$$\frac{-6a^{2}-12ab+8b^{2}}{\left(a+b\right)\left(-a+b\right)}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+b\right)\left(-a+b\right) and a^{2}+b^{2} is \left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right). Multiply \frac{-6a^{2}-12ab+8b^{2}}{\left(a+b\right)\left(-a+b\right)} times \frac{a^{2}+b^{2}}{a^{2}+b^{2}}. Multiply \frac{4b^{2}-6a^{2}}{a^{2}+b^{2}} times \frac{\left(a+b\right)\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)}.

$$\frac{\left(-6a^{2}-12ab+8b^{2}\right)\left(a^{2}+b^{2}\right)}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}+\frac{\left(4b^{2}-6a^{2}\right)\left(a+b\right)\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Since \frac{\left(-6a^{2}-12ab+8b^{2}\right)\left(a^{2}+b^{2}\right)}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)} and \frac{\left(4b^{2}-6a^{2}\right)\left(a+b\right)\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(-6a^{2}-12ab+8b^{2}\right)\left(a^{2}+b^{2}\right)+\left(4b^{2}-6a^{2}\right)\left(a+b\right)\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Do the multiplications in \left(-6a^{2}-12ab+8b^{2}\right)\left(a^{2}+b^{2}\right)+\left(4b^{2}-6a^{2}\right)\left(a+b\right)\left(-a+b\right).

$$\frac{-6a^{4}-6a^{2}b^{2}-12a^{3}b-12ab^{3}+8b^{2}a^{2}+8b^{4}-4b^{2}a^{2}+4b^{4}+6a^{4}-6a^{2}b^{2}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Combine like terms in -6a^{4}-6a^{2}b^{2}-12a^{3}b-12ab^{3}+8b^{2}a^{2}+8b^{4}-4b^{2}a^{2}+4b^{4}+6a^{4}-6a^{2}b^{2}.

$$\frac{-8a^{2}b^{2}-12ab^{3}-12a^{3}b+12b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Factor b^{4}-a^{4}.

$$\frac{-8a^{2}b^{2}-12ab^{3}-12a^{3}b+12b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}-\frac{20b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}$$

Since \frac{-8a^{2}b^{2}-12ab^{3}-12a^{3}b+12b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)} and \frac{20b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-8a^{2}b^{2}-12ab^{3}-12a^{3}b+12b^{4}-20b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}$$

Combine like terms in -8a^{2}b^{2}-12ab^{3}-12a^{3}b+12b^{4}-20b^{4}.

$$\frac{-8a^{2}b^{2}-12a^{3}b-12ab^{3}-8b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}$$

Factor the expressions that are not already factored in \frac{-8a^{2}b^{2}-12a^{3}b-12ab^{3}-8b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}.

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

Cancel out a^{2}+b^{2} in both numerator and denominator.

$$\frac{4b\left(-3a-2b\right)}{\left(a+b\right)\left(-a+b\right)}$$

Expand \left(a+b\right)\left(-a+b\right).

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

Use the distributive property to multiply 4b by -3a-2b.

$$\frac{-12ba-8b^{2}}{-a^{2}+b^{2}}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+b and a-b is \left(a+b\right)\left(a-b\right). Multiply \frac{4a+6b}{a+b} times \frac{a-b}{a-b}. Multiply \frac{6a+4b}{a-b} times \frac{a+b}{a+b}.

$$\frac{\left(4a+6b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}+\frac{\left(6a+4b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}+\frac{4a^{2}+6b^{2}}{b^{2}-a^{2}}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Since \frac{\left(4a+6b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)} and \frac{\left(6a+4b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4a+6b\right)\left(a-b\right)+\left(6a+4b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}+\frac{4a^{2}+6b^{2}}{b^{2}-a^{2}}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Do the multiplications in \left(4a+6b\right)\left(a-b\right)+\left(6a+4b\right)\left(a+b\right).

$$\frac{4a^{2}-4ab+6ba-6b^{2}+6a^{2}+6ba+4ba+4b^{2}}{\left(a+b\right)\left(a-b\right)}+\frac{4a^{2}+6b^{2}}{b^{2}-a^{2}}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Combine like terms in 4a^{2}-4ab+6ba-6b^{2}+6a^{2}+6ba+4ba+4b^{2}.

$$\frac{10a^{2}-2b^{2}+12ab}{\left(a+b\right)\left(a-b\right)}+\frac{4a^{2}+6b^{2}}{b^{2}-a^{2}}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Factor b^{2}-a^{2}.

$$\frac{10a^{2}-2b^{2}+12ab}{\left(a+b\right)\left(a-b\right)}+\frac{4a^{2}+6b^{2}}{\left(a+b\right)\left(-a+b\right)}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+b\right)\left(a-b\right) and \left(a+b\right)\left(-a+b\right) is \left(a+b\right)\left(-a+b\right). Multiply \frac{10a^{2}-2b^{2}+12ab}{\left(a+b\right)\left(a-b\right)} times \frac{-1}{-1}.

$$\frac{-\left(10a^{2}-2b^{2}+12ab\right)}{\left(a+b\right)\left(-a+b\right)}+\frac{4a^{2}+6b^{2}}{\left(a+b\right)\left(-a+b\right)}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Since \frac{-\left(10a^{2}-2b^{2}+12ab\right)}{\left(a+b\right)\left(-a+b\right)} and \frac{4a^{2}+6b^{2}}{\left(a+b\right)\left(-a+b\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(10a^{2}-2b^{2}+12ab\right)+4a^{2}+6b^{2}}{\left(a+b\right)\left(-a+b\right)}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Do the multiplications in -\left(10a^{2}-2b^{2}+12ab\right)+4a^{2}+6b^{2}.

$$\frac{-10a^{2}+2b^{2}-12ab+4a^{2}+6b^{2}}{\left(a+b\right)\left(-a+b\right)}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Combine like terms in -10a^{2}+2b^{2}-12ab+4a^{2}+6b^{2}.

$$\frac{-6a^{2}-12ab+8b^{2}}{\left(a+b\right)\left(-a+b\right)}+\frac{4b^{2}-6a^{2}}{a^{2}+b^{2}}-\frac{20b^{4}}{b^{4}-a^{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+b\right)\left(-a+b\right) and a^{2}+b^{2} is \left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right). Multiply \frac{-6a^{2}-12ab+8b^{2}}{\left(a+b\right)\left(-a+b\right)} times \frac{a^{2}+b^{2}}{a^{2}+b^{2}}. Multiply \frac{4b^{2}-6a^{2}}{a^{2}+b^{2}} times \frac{\left(a+b\right)\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)}.

$$\frac{\left(-6a^{2}-12ab+8b^{2}\right)\left(a^{2}+b^{2}\right)}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}+\frac{\left(4b^{2}-6a^{2}\right)\left(a+b\right)\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Since \frac{\left(-6a^{2}-12ab+8b^{2}\right)\left(a^{2}+b^{2}\right)}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)} and \frac{\left(4b^{2}-6a^{2}\right)\left(a+b\right)\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(-6a^{2}-12ab+8b^{2}\right)\left(a^{2}+b^{2}\right)+\left(4b^{2}-6a^{2}\right)\left(a+b\right)\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Do the multiplications in \left(-6a^{2}-12ab+8b^{2}\right)\left(a^{2}+b^{2}\right)+\left(4b^{2}-6a^{2}\right)\left(a+b\right)\left(-a+b\right).

$$\frac{-6a^{4}-6a^{2}b^{2}-12a^{3}b-12ab^{3}+8b^{2}a^{2}+8b^{4}-4b^{2}a^{2}+4b^{4}+6a^{4}-6a^{2}b^{2}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Combine like terms in -6a^{4}-6a^{2}b^{2}-12a^{3}b-12ab^{3}+8b^{2}a^{2}+8b^{4}-4b^{2}a^{2}+4b^{4}+6a^{4}-6a^{2}b^{2}.

$$\frac{-8a^{2}b^{2}-12ab^{3}-12a^{3}b+12b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}-\frac{20b^{4}}{b^{4}-a^{4}}$$

Factor b^{4}-a^{4}.

$$\frac{-8a^{2}b^{2}-12ab^{3}-12a^{3}b+12b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}-\frac{20b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}$$

Since \frac{-8a^{2}b^{2}-12ab^{3}-12a^{3}b+12b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)} and \frac{20b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-8a^{2}b^{2}-12ab^{3}-12a^{3}b+12b^{4}-20b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}$$

Combine like terms in -8a^{2}b^{2}-12ab^{3}-12a^{3}b+12b^{4}-20b^{4}.

$$\frac{-8a^{2}b^{2}-12a^{3}b-12ab^{3}-8b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}$$

Factor the expressions that are not already factored in \frac{-8a^{2}b^{2}-12a^{3}b-12ab^{3}-8b^{4}}{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}.

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

Cancel out a^{2}+b^{2} in both numerator and denominator.

$$\frac{4b\left(-3a-2b\right)}{\left(a+b\right)\left(-a+b\right)}$$

Expand \left(a+b\right)\left(-a+b\right).

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

Use the distributive property to multiply 4b by -3a-2b.

$$\frac{-12ba-8b^{2}}{-a^{2}+b^{2}}$$