Basic Math Examples

$(a + b) (4 a - 3 b) - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Apply the distributive property.

$a (4 a - 3 b) + b (4 a - 3 b) - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.1.2

Apply the distributive property.

$a (4 a) + a (- 3 b) + b (4 a - 3 b) - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.1.3

Apply the distributive property.

$a (4 a) + a (- 3 b) + b (4 a) + b (- 3 b) - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$4 a \cdot a + a (- 3 b) + b (4 a) + b (- 3 b) - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.2.1.2

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

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Step 1.2.1.2.1

Move $a$ .

$4 (a \cdot a) + a (- 3 b) + b (4 a) + b (- 3 b) - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.2.1.2.2

Multiply $a$ by $a$ .

$4 a^{2} + a (- 3 b) + b (4 a) + b (- 3 b) - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.2.1.3

Rewrite using the commutative property of multiplication.

$4 a^{2} - 3 a b + b (4 a) + b (- 3 b) - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.2.1.4

Rewrite using the commutative property of multiplication.

$4 a^{2} - 3 a b + 4 b a + b (- 3 b) - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.2.1.5

Rewrite using the commutative property of multiplication.

$4 a^{2} - 3 a b + 4 b a - 3 b \cdot b - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.2.1.6

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

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Step 1.2.1.6.1

Move $b$ .

$4 a^{2} - 3 a b + 4 b a - 3 (b \cdot b) - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.2.1.6.2

Multiply $b$ by $b$ .

$4 a^{2} - 3 a b + 4 b a - 3 b^{2} - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.2.2

Add $- 3 a b$ and $4 b a$ .

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Step 1.2.2.1

Move $b$ .

$4 a^{2} - 3 a b + 4 a b - 3 b^{2} - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.2.2.2

Add $- 3 a b$ and $4 a b$ .

$4 a^{2} + 1 a b - 3 b^{2} - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.3

Multiply $a$ by $1$ .

$4 a^{2} + a b - 3 b^{2} - (5 a - 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.4

Apply the distributive property.

$4 a^{2} + a b - 3 b^{2} + (- (5 a) - (- 2 b)) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.5

Multiply $5$ by $- 1$ .

$4 a^{2} + a b - 3 b^{2} + (- 5 a - (- 2 b)) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.6

Multiply $- 2$ by $- 1$ .

$4 a^{2} + a b - 3 b^{2} + (- 5 a + 2 b) (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.7

Expand $(- 5 a + 2 b) (3 a + b)$ using the FOIL Method.

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Step 1.7.1

Apply the distributive property.

$4 a^{2} + a b - 3 b^{2} - 5 a (3 a + b) + 2 b (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.7.2

Apply the distributive property.

$4 a^{2} + a b - 3 b^{2} - 5 a (3 a) - 5 a b + 2 b (3 a + b) - (a + b) (3 a + 6 b)$

Step 1.7.3

Apply the distributive property.

$4 a^{2} + a b - 3 b^{2} - 5 a (3 a) - 5 a b + 2 b (3 a) + 2 b \cdot b - (a + b) (3 a + 6 b)$

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Rewrite using the commutative property of multiplication.

$4 a^{2} + a b - 3 b^{2} - 5 \cdot 3 a \cdot a - 5 a b + 2 b (3 a) + 2 b \cdot b - (a + b) (3 a + 6 b)$

Step 1.8.1.2

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

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Step 1.8.1.2.1

Move $a$ .

$4 a^{2} + a b - 3 b^{2} - 5 \cdot 3 (a \cdot a) - 5 a b + 2 b (3 a) + 2 b \cdot b - (a + b) (3 a + 6 b)$

Step 1.8.1.2.2

Multiply $a$ by $a$ .

$4 a^{2} + a b - 3 b^{2} - 5 \cdot 3 a^{2} - 5 a b + 2 b (3 a) + 2 b \cdot b - (a + b) (3 a + 6 b)$

Step 1.8.1.3

Multiply $- 5$ by $3$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} - 5 a b + 2 b (3 a) + 2 b \cdot b - (a + b) (3 a + 6 b)$

Step 1.8.1.4

Rewrite using the commutative property of multiplication.

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} - 5 a b + 2 \cdot 3 b a + 2 b \cdot b - (a + b) (3 a + 6 b)$

Step 1.8.1.5

Multiply $2$ by $3$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} - 5 a b + 6 b a + 2 b \cdot b - (a + b) (3 a + 6 b)$

Step 1.8.1.6

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

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Step 1.8.1.6.1

Move $b$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} - 5 a b + 6 b a + 2 (b \cdot b) - (a + b) (3 a + 6 b)$

Step 1.8.1.6.2

Multiply $b$ by $b$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} - 5 a b + 6 b a + 2 b^{2} - (a + b) (3 a + 6 b)$

Step 1.8.2

Add $- 5 a b$ and $6 b a$ .

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Step 1.8.2.1

Move $b$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} - 5 a b + 6 a b + 2 b^{2} - (a + b) (3 a + 6 b)$

Step 1.8.2.2

Add $- 5 a b$ and $6 a b$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + 1 a b + 2 b^{2} - (a + b) (3 a + 6 b)$

Step 1.9

Multiply $a$ by $1$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - (a + b) (3 a + 6 b)$

Step 1.10

Apply the distributive property.

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} + (- a - b) (3 a + 6 b)$

Step 1.11

Expand $(- a - b) (3 a + 6 b)$ using the FOIL Method.

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Step 1.11.1

Apply the distributive property.

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - a (3 a + 6 b) - b (3 a + 6 b)$

Step 1.11.2

Apply the distributive property.

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - a (3 a) - a (6 b) - b (3 a + 6 b)$

Step 1.11.3

Apply the distributive property.

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - a (3 a) - a (6 b) - b (3 a) - b (6 b)$

Step 1.12

Simplify and combine like terms.

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Step 1.12.1

Simplify each term.

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Step 1.12.1.1

Rewrite using the commutative property of multiplication.

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 1 \cdot 3 a \cdot a - a (6 b) - b (3 a) - b (6 b)$

Step 1.12.1.2

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

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Step 1.12.1.2.1

Move $a$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 1 \cdot 3 (a \cdot a) - a (6 b) - b (3 a) - b (6 b)$

Step 1.12.1.2.2

Multiply $a$ by $a$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 1 \cdot 3 a^{2} - a (6 b) - b (3 a) - b (6 b)$

Step 1.12.1.3

Multiply $- 1$ by $3$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - a (6 b) - b (3 a) - b (6 b)$

Step 1.12.1.4

Rewrite using the commutative property of multiplication.

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - 1 \cdot 6 a b - b (3 a) - b (6 b)$

Step 1.12.1.5

Multiply $- 1$ by $6$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - 6 a b - b (3 a) - b (6 b)$

Step 1.12.1.6

Rewrite using the commutative property of multiplication.

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - 6 a b - 1 \cdot 3 b a - b (6 b)$

Step 1.12.1.7

Multiply $- 1$ by $3$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - 6 a b - 3 b a - b (6 b)$

Step 1.12.1.8

Rewrite using the commutative property of multiplication.

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - 6 a b - 3 b a - 1 \cdot 6 b \cdot b$

Step 1.12.1.9

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

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Step 1.12.1.9.1

Move $b$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - 6 a b - 3 b a - 1 \cdot 6 (b \cdot b)$

Step 1.12.1.9.2

Multiply $b$ by $b$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - 6 a b - 3 b a - 1 \cdot 6 b^{2}$

Step 1.12.1.10

Multiply $- 1$ by $6$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - 6 a b - 3 b a - 6 b^{2}$

Step 1.12.2

Subtract $3 b a$ from $- 6 a b$ .

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Step 1.12.2.1

Move $b$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - 6 a b - 3 a b - 6 b^{2}$

Step 1.12.2.2

Subtract $3 a b$ from $- 6 a b$ .

$4 a^{2} + a b - 3 b^{2} - 15 a^{2} + a b + 2 b^{2} - 3 a^{2} - 9 a b - 6 b^{2}$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $15 a^{2}$ from $4 a^{2}$ .

$- 11 a^{2} + a b - 3 b^{2} + a b + 2 b^{2} - 3 a^{2} - 9 a b - 6 b^{2}$

Step 2.2

Subtract $3 a^{2}$ from $- 11 a^{2}$ .

$- 14 a^{2} + a b - 3 b^{2} + a b + 2 b^{2} - 9 a b - 6 b^{2}$

Step 2.3

Add $a b$ and $a b$ .

$- 14 a^{2} - 3 b^{2} + 2 a b + 2 b^{2} - 9 a b - 6 b^{2}$

Step 2.4

Add $- 3 b^{2}$ and $2 b^{2}$ .

$- 14 a^{2} - b^{2} + 2 a b - 9 a b - 6 b^{2}$

Step 2.5

Subtract $6 b^{2}$ from $- b^{2}$ .

$- 14 a^{2} - 7 b^{2} + 2 a b - 9 a b$

Step 2.6

Subtract $9 a b$ from $2 a b$ .

$- 14 a^{2} - 7 b^{2} - 7 a b$