Simplify (a+2b)(a^2-2ab-4b^2)

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

Step 1

Expand $(a + 2 b) (a^{2} - 2 a b - 4 b^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

Simplify each term.

Multiply $a$ by $a^{2}$ by adding the exponents.

Multiply $a$ by $a^{2}$ .

Raise $a$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $a$ .

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

Multiply $a$ by $a$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $b$ .

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

Multiply $b$ by $b$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $2$ by $- 2$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $b$ by $b^{2}$ by adding the exponents.

Move $b^{2}$ .

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

Multiply $b^{2}$ by $b$ .

Raise $b$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $2$ and $1$ .

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

Multiply $2$ by $- 4$ .

$a^{3} - 2 a^{2} b - 4 a b^{2} + 2 b a^{2} - 4 b^{2} a - 8 b^{3}$

Combine the opposite terms in $a^{3} - 2 a^{2} b - 4 a b^{2} + 2 b a^{2} - 4 b^{2} a - 8 b^{3}$ .

Reorder the factors in the terms $- 2 a^{2} b$ and $2 b a^{2}$ .

$a^{3} - 2 a^{2} b - 4 a b^{2} + 2 a^{2} b - 4 b^{2} a - 8 b^{3}$

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

$a^{3} - 4 a b^{2} + 0 - 4 b^{2} a - 8 b^{3}$

Add $a^{3} - 4 a b^{2}$ and $0$ .

$a^{3} - 4 a b^{2} - 4 b^{2} a - 8 b^{3}$

Step 3

Subtract $4 b^{2} a$ from $- 4 a b^{2}$ .

Move $b^{2}$ .

$a^{3} - 4 a b^{2} - 4 a b^{2} - 8 b^{3}$

Subtract $4 a b^{2}$ from $- 4 a b^{2}$ .

$a^{3} - 8 a b^{2} - 8 b^{3}$

Simplify (a+2b)(a^2-2ab-4b^2)

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