Calculus Examples

$\frac{10 a + 3 b}{10 a^{2} + 3 a b - 4 b^{2}} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Factor by grouping.

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Step 1.1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 10 \cdot - 4 = - 40$ and whose sum is $b = 3$ .

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Step 1.1.1.1.1

Reorder terms.

$\frac{10 a + 3 b}{10 a^{2} - 4 b^{2} + 3 a b} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1.1.1.1.2

Reorder $- 4 b^{2}$ and $3 a b$ .

$\frac{10 a + 3 b}{10 a^{2} + 3 a b - 4 b^{2}} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1.1.1.1.3

Factor $3$ out of $3 a b$ .

$\frac{10 a + 3 b}{10 a^{2} + 3 (a b) - 4 b^{2}} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1.1.1.1.4

Rewrite $3$ as $- 5$ plus $8$

$\frac{10 a + 3 b}{10 a^{2} + (- 5 + 8) (a b) - 4 b^{2}} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1.1.1.1.5

Apply the distributive property.

$\frac{10 a + 3 b}{10 a^{2} - 5 (a b) + 8 (a b) - 4 b^{2}} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1.1.1.1.6

Move parentheses.

$\frac{10 a + 3 b}{10 a^{2} - 5 a b + 8 a b - 4 b^{2}} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1.1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.1.2.1

Group the first two terms and the last two terms.

$\frac{10 a + 3 b}{(10 a^{2} - 5 a b) + 8 a b - 4 b^{2}} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1.1.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{10 a + 3 b}{5 a (2 a - 1 b) + 4 b (2 a - b)} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1.1.1.3

Factor the polynomial by factoring out the greatest common factor, $2 a - 1 b$ .

$\frac{10 a + 3 b}{(2 a - 1 b) (5 a + 4 b)} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1.1.2

Rewrite $- 1 b$ as $- b$ .

$\frac{10 a + 3 b}{(2 a - b) (5 a + 4 b)} - \frac{a + 2 b}{25 a^{2} - 16 b^{2}}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $25 a^{2}$ as ${(5 a)}^{2}$ .

$\frac{10 a + 3 b}{(2 a - b) (5 a + 4 b)} - \frac{a + 2 b}{{(5 a)}^{2} - 16 b^{2}}$

Step 1.2.2

Rewrite $16 b^{2}$ as ${(4 b)}^{2}$ .

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

Step 1.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 a$ and $b = 4 b$ .

$\frac{10 a + 3 b}{(2 a - b) (5 a + 4 b)} - \frac{a + 2 b}{(5 a + 4 b) (5 a - (4 b))}$

Step 1.2.4

Multiply $4$ by $- 1$ .

$\frac{10 a + 3 b}{(2 a - b) (5 a + 4 b)} - \frac{a + 2 b}{(5 a + 4 b) (5 a - 4 b)}$

Step 2

To write $\frac{10 a + 3 b}{(2 a - b) (5 a + 4 b)}$ as a fraction with a common denominator, multiply by $\frac{5 a - 4 b}{5 a - 4 b}$ .

$\frac{10 a + 3 b}{(2 a - b) (5 a + 4 b)} \cdot \frac{5 a - 4 b}{5 a - 4 b} - \frac{a + 2 b}{(5 a + 4 b) (5 a - 4 b)}$

Step 3

To write $- \frac{a + 2 b}{(5 a + 4 b) (5 a - 4 b)}$ as a fraction with a common denominator, multiply by $\frac{2 a - b}{2 a - b}$ .

$\frac{10 a + 3 b}{(2 a - b) (5 a + 4 b)} \cdot \frac{5 a - 4 b}{5 a - 4 b} - \frac{a + 2 b}{(5 a + 4 b) (5 a - 4 b)} \cdot \frac{2 a - b}{2 a - b}$

Step 4

Write each expression with a common denominator of $(2 a - b) (5 a + 4 b) (5 a - 4 b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{10 a + 3 b}{(2 a - b) (5 a + 4 b)}$ by $\frac{5 a - 4 b}{5 a - 4 b}$ .

$\frac{(10 a + 3 b) (5 a - 4 b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)} - \frac{a + 2 b}{(5 a + 4 b) (5 a - 4 b)} \cdot \frac{2 a - b}{2 a - b}$

Step 4.2

Multiply $\frac{a + 2 b}{(5 a + 4 b) (5 a - 4 b)}$ by $\frac{2 a - b}{2 a - b}$ .

$\frac{(10 a + 3 b) (5 a - 4 b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)} - \frac{(a + 2 b) (2 a - b)}{(5 a + 4 b) (5 a - 4 b) (2 a - b)}$

Step 4.3

Reorder the factors of $(5 a + 4 b) (5 a - 4 b) (2 a - b)$ .

$\frac{(10 a + 3 b) (5 a - 4 b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)} - \frac{(a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(10 a + 3 b) (5 a - 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(10 a + 3 b) (5 a - 4 b)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{10 a (5 a - 4 b) + 3 b (5 a - 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.1.2

Apply the distributive property.

$\frac{10 a (5 a) + 10 a (- 4 b) + 3 b (5 a - 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.1.3

Apply the distributive property.

$\frac{10 a (5 a) + 10 a (- 4 b) + 3 b (5 a) + 3 b (- 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{10 \cdot 5 a \cdot a + 10 a (- 4 b) + 3 b (5 a) + 3 b (- 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 6.2.1.2.1

Move $a$ .

$\frac{10 \cdot 5 (a \cdot a) + 10 a (- 4 b) + 3 b (5 a) + 3 b (- 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.2.2

Multiply $a$ by $a$ .

$\frac{10 \cdot 5 a^{2} + 10 a (- 4 b) + 3 b (5 a) + 3 b (- 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.3

Multiply $10$ by $5$ .

$\frac{50 a^{2} + 10 a (- 4 b) + 3 b (5 a) + 3 b (- 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{50 a^{2} + 10 \cdot - 4 a b + 3 b (5 a) + 3 b (- 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.5

Multiply $10$ by $- 4$ .

$\frac{50 a^{2} - 40 a b + 3 b (5 a) + 3 b (- 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.6

Rewrite using the commutative property of multiplication.

$\frac{50 a^{2} - 40 a b + 3 \cdot 5 b a + 3 b (- 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.7

Multiply $3$ by $5$ .

$\frac{50 a^{2} - 40 a b + 15 b a + 3 b (- 4 b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.8

Rewrite using the commutative property of multiplication.

$\frac{50 a^{2} - 40 a b + 15 b a + 3 \cdot - 4 b \cdot b - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.9

Multiply $b$ by $b$ by adding the exponents.

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Step 6.2.1.9.1

Move $b$ .

$\frac{50 a^{2} - 40 a b + 15 b a + 3 \cdot - 4 (b \cdot b) - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.9.2

Multiply $b$ by $b$ .

$\frac{50 a^{2} - 40 a b + 15 b a + 3 \cdot - 4 b^{2} - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.1.10

Multiply $3$ by $- 4$ .

$\frac{50 a^{2} - 40 a b + 15 b a - 12 b^{2} - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.2

Add $- 40 a b$ and $15 b a$ .

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Step 6.2.2.1

Move $b$ .

$\frac{50 a^{2} - 40 a b + 15 a b - 12 b^{2} - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.2.2.2

Add $- 40 a b$ and $15 a b$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - (a + 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.3

Apply the distributive property.

$\frac{50 a^{2} - 25 a b - 12 b^{2} + (- a - (2 b)) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.4

Multiply $2$ by $- 1$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} + (- a - 2 b) (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.5

Expand $(- a - 2 b) (2 a - b)$ using the FOIL Method.

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Step 6.5.1

Apply the distributive property.

$\frac{50 a^{2} - 25 a b - 12 b^{2} - a (2 a - b) - 2 b (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.5.2

Apply the distributive property.

$\frac{50 a^{2} - 25 a b - 12 b^{2} - a (2 a) - a (- b) - 2 b (2 a - b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.5.3

Apply the distributive property.

$\frac{50 a^{2} - 25 a b - 12 b^{2} - a (2 a) - a (- b) - 2 b (2 a) - 2 b (- b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6

Simplify and combine like terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

Rewrite using the commutative property of multiplication.

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 1 \cdot 2 a \cdot a - a (- b) - 2 b (2 a) - 2 b (- b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 6.6.1.2.1

Move $a$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 1 \cdot 2 (a \cdot a) - a (- b) - 2 b (2 a) - 2 b (- b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.2.2

Multiply $a$ by $a$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 1 \cdot 2 a^{2} - a (- b) - 2 b (2 a) - 2 b (- b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.3

Multiply $- 1$ by $2$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} - a (- b) - 2 b (2 a) - 2 b (- b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.4

Rewrite using the commutative property of multiplication.

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} - 1 \cdot - 1 a b - 2 b (2 a) - 2 b (- b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.5

Multiply $- 1$ by $- 1$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} + 1 a b - 2 b (2 a) - 2 b (- b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.6

Multiply $a$ by $1$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} + a b - 2 b (2 a) - 2 b (- b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.7

Rewrite using the commutative property of multiplication.

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} + a b - 2 \cdot 2 b a - 2 b (- b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.8

Multiply $- 2$ by $2$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} + a b - 4 b a - 2 b (- b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.9

Rewrite using the commutative property of multiplication.

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} + a b - 4 b a - 2 \cdot - 1 b \cdot b}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.10

Multiply $b$ by $b$ by adding the exponents.

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Step 6.6.1.10.1

Move $b$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} + a b - 4 b a - 2 \cdot - 1 (b \cdot b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.10.2

Multiply $b$ by $b$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} + a b - 4 b a - 2 \cdot - 1 b^{2}}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.1.11

Multiply $- 2$ by $- 1$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} + a b - 4 b a + 2 b^{2}}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.2

Subtract $4 b a$ from $a b$ .

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Step 6.6.2.1

Move $b$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} + a b - 4 a b + 2 b^{2}}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.6.2.2

Subtract $4 a b$ from $a b$ .

$\frac{50 a^{2} - 25 a b - 12 b^{2} - 2 a^{2} - 3 a b + 2 b^{2}}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.7

Subtract $2 a^{2}$ from $50 a^{2}$ .

$\frac{48 a^{2} - 25 a b - 12 b^{2} - 3 a b + 2 b^{2}}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.8

Subtract $3 a b$ from $- 25 a b$ .

$\frac{48 a^{2} - 12 b^{2} - 28 a b + 2 b^{2}}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.9

Add $- 12 b^{2}$ and $2 b^{2}$ .

$\frac{48 a^{2} - 10 b^{2} - 28 a b}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10

Rewrite $48 a^{2} - 10 b^{2} - 28 a b$ in a factored form.

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Step 6.10.1

Factor $2$ out of $48 a^{2} - 10 b^{2} - 28 a b$ .

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Step 6.10.1.1

Factor $2$ out of $48 a^{2}$ .

$\frac{2 (24 a^{2}) - 10 b^{2} - 28 a b}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.1.2

Factor $2$ out of $- 10 b^{2}$ .

$\frac{2 (24 a^{2}) + 2 (- 5 b^{2}) - 28 a b}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.1.3

Factor $2$ out of $- 28 a b$ .

$\frac{2 (24 a^{2}) + 2 (- 5 b^{2}) + 2 (- 14 a b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.1.4

Factor $2$ out of $2 (24 a^{2}) + 2 (- 5 b^{2})$ .

$\frac{2 (24 a^{2} - 5 b^{2}) + 2 (- 14 a b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.1.5

Factor $2$ out of $2 (24 a^{2} - 5 b^{2}) + 2 (- 14 a b)$ .

$\frac{2 (24 a^{2} - 5 b^{2} - 14 a b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.2

Factor by grouping.

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Step 6.10.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 24 \cdot - 5 = - 120$ and whose sum is $b = - 14$ .

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Step 6.10.2.1.1

Reorder $- 5 b^{2}$ and $- 14 a b$ .

$\frac{2 (24 a^{2} - 14 a b - 5 b^{2})}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.2.1.2

Factor $- 14$ out of $- 14 a b$ .

$\frac{2 (24 a^{2} - 14 (a b) - 5 b^{2})}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.2.1.3

Rewrite $- 14$ as $6$ plus $- 20$

$\frac{2 (24 a^{2} + (6 - 20) (a b) - 5 b^{2})}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.2.1.4

Apply the distributive property.

$\frac{2 (24 a^{2} + 6 (a b) - 20 (a b) - 5 b^{2})}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.2.1.5

Move parentheses.

$\frac{2 (24 a^{2} + 6 a b - 20 a b - 5 b^{2})}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.2.2

Factor out the greatest common factor from each group.

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Step 6.10.2.2.1

Group the first two terms and the last two terms.

$\frac{2 ((24 a^{2} + 6 a b) - 20 a b - 5 b^{2})}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 (6 a (4 a + b) - 5 b (4 a + b))}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

Step 6.10.2.3

Factor the polynomial by factoring out the greatest common factor, $4 a + b$ .

$\frac{2 ((4 a + b) (6 a - 5 b))}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$

$\frac{2 (4 a + b) (6 a - 5 b)}{(2 a - b) (5 a + 4 b) (5 a - 4 b)}$