Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

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

Step 1.2

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

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Step 1.2.1

Move $a$ .

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

Step 1.2.2

Multiply $a$ by $a$ .

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

Step 1.3

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

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Step 1.3.1

Move $b$ .

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

Step 1.3.2

Multiply $b$ by $b$ .

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

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $4$ by $3$ .

$12 a^{2} b + 3 a b^{2} \cdot - 5 + 4 b^{2} (2 a^{2} + 1)$

Step 1.4.2

Multiply $- 5$ by $3$ .

$12 a^{2} b - 15 a b^{2} + 4 b^{2} (2 a^{2} + 1)$

Step 1.5

Apply the distributive property.

$12 a^{2} b - 15 a b^{2} + 4 b^{2} (2 a^{2}) + 4 b^{2} \cdot 1$

Step 1.6

Rewrite using the commutative property of multiplication.

$12 a^{2} b - 15 a b^{2} + 4 \cdot 2 b^{2} a^{2} + 4 b^{2} \cdot 1$

Step 1.7

Multiply $4$ by $1$ .

$12 a^{2} b - 15 a b^{2} + 4 \cdot 2 b^{2} a^{2} + 4 b^{2}$

Step 1.8

Multiply $4$ by $2$ .

$12 a^{2} b - 15 a b^{2} + 8 b^{2} a^{2} + 4 b^{2}$

Step 2

Simplify the expression.

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Step 2.1

Move $b^{2}$ .

$12 a^{2} b - 15 a b^{2} + 8 a^{2} b^{2} + 4 b^{2}$

Step 2.2

Move $- 15 a b^{2}$ .

$12 a^{2} b + 8 a^{2} b^{2} - 15 a b^{2} + 4 b^{2}$

Step 2.3

Reorder $12 a^{2} b$ and $8 a^{2} b^{2}$ .

$8 a^{2} b^{2} + 12 a^{2} b - 15 a b^{2} + 4 b^{2}$