Basic Math Examples

$4 p {(p + 2 q)}^{2} - 12 q (2 q + p)$

Step 1

Factor $4$ out of $4 p {(p + 2 q)}^{2} - 12 q (2 q + p)$ .

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Step 1.1

Factor $4$ out of $4 p {(p + 2 q)}^{2}$ .

$4 (p {(p + 2 q)}^{2}) - 12 q (2 q + p)$

Step 1.2

Factor $4$ out of $- 12 q (2 q + p)$ .

$4 (p {(p + 2 q)}^{2}) + 4 (- 3 q (2 q + p))$

Step 1.3

Factor $4$ out of $4 (p {(p + 2 q)}^{2}) + 4 (- 3 q (2 q + p))$ .

$4 (p {(p + 2 q)}^{2} - 3 q (2 q + p))$

Step 2

Rewrite ${(p + 2 q)}^{2}$ as $(p + 2 q) (p + 2 q)$ .

$4 (p ((p + 2 q) (p + 2 q)) - 3 q (2 q + p))$

Step 3

Expand $(p + 2 q) (p + 2 q)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$4 (p (p (p + 2 q) + 2 q (p + 2 q)) - 3 q (2 q + p))$

Step 3.2

Apply the distributive property.

$4 (p (p \cdot p + p (2 q) + 2 q (p + 2 q)) - 3 q (2 q + p))$

Step 3.3

Apply the distributive property.

$4 (p (p \cdot p + p (2 q) + 2 q p + 2 q (2 q)) - 3 q (2 q + p))$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $p$ by $p$ .

$4 (p (p^{2} + p (2 q) + 2 q p + 2 q (2 q)) - 3 q (2 q + p))$

Step 4.1.2

Rewrite using the commutative property of multiplication.

$4 (p (p^{2} + 2 p q + 2 q p + 2 q (2 q)) - 3 q (2 q + p))$

Step 4.1.3

Rewrite using the commutative property of multiplication.

$4 (p (p^{2} + 2 p q + 2 q p + 2 \cdot 2 q \cdot q) - 3 q (2 q + p))$

Step 4.1.4

Multiply $q$ by $q$ by adding the exponents.

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Step 4.1.4.1

Move $q$ .

$4 (p (p^{2} + 2 p q + 2 q p + 2 \cdot 2 (q \cdot q)) - 3 q (2 q + p))$

Step 4.1.4.2

Multiply $q$ by $q$ .

$4 (p (p^{2} + 2 p q + 2 q p + 2 \cdot 2 q^{2}) - 3 q (2 q + p))$

Step 4.1.5

Multiply $2$ by $2$ .

$4 (p (p^{2} + 2 p q + 2 q p + 4 q^{2}) - 3 q (2 q + p))$

Step 4.2

Add $2 p q$ and $2 q p$ .

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Step 4.2.1

Move $q$ .

$4 (p (p^{2} + 2 p q + 2 p q + 4 q^{2}) - 3 q (2 q + p))$

Step 4.2.2

Add $2 p q$ and $2 p q$ .

$4 (p (p^{2} + 4 p q + 4 q^{2}) - 3 q (2 q + p))$

Step 5

Apply the distributive property.

$4 (p \cdot p^{2} + p (4 p q) + p (4 q^{2}) - 3 q (2 q + p))$

Step 6

Simplify.

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Step 6.1

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

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Step 6.1.1

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

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Step 6.1.1.1

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

$4 (p^{1} p^{2} + p (4 p q) + p (4 q^{2}) - 3 q (2 q + p))$

Step 6.1.1.2

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

$4 (p^{1 + 2} + p (4 p q) + p (4 q^{2}) - 3 q (2 q + p))$

Step 6.1.2

Add $1$ and $2$ .

$4 (p^{3} + p (4 p q) + p (4 q^{2}) - 3 q (2 q + p))$

Step 6.2

Rewrite using the commutative property of multiplication.

$4 (p^{3} + 4 p (p q) + p (4 q^{2}) - 3 q (2 q + p))$

Step 6.3

Rewrite using the commutative property of multiplication.

$4 (p^{3} + 4 p (p q) + 4 p q^{2} - 3 q (2 q + p))$

Step 7

Multiply $p$ by $p$ by adding the exponents.

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Step 7.1

Move $p$ .

$4 (p^{3} + 4 (p \cdot p) q + 4 p q^{2} - 3 q (2 q + p))$

Step 7.2

Multiply $p$ by $p$ .

$4 (p^{3} + 4 p^{2} q + 4 p q^{2} - 3 q (2 q + p))$

Step 8

Apply the distributive property.

$4 (p^{3} + 4 p^{2} q + 4 p q^{2} - 3 q (2 q) - 3 q p)$

Step 9

Rewrite using the commutative property of multiplication.

$4 (p^{3} + 4 p^{2} q + 4 p q^{2} - 3 \cdot 2 q \cdot q - 3 q p)$

Step 10

Simplify each term.

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Step 10.1

Multiply $q$ by $q$ by adding the exponents.

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Step 10.1.1

Move $q$ .

$4 (p^{3} + 4 p^{2} q + 4 p q^{2} - 3 \cdot 2 (q \cdot q) - 3 q p)$

Step 10.1.2

Multiply $q$ by $q$ .

$4 (p^{3} + 4 p^{2} q + 4 p q^{2} - 3 \cdot 2 q^{2} - 3 q p)$

Step 10.2

Multiply $- 3$ by $2$ .

$4 (p^{3} + 4 p^{2} q + 4 p q^{2} - 6 q^{2} - 3 q p)$