Pre-Algebra Examples

$(3 m^{3} + m^{2} - 2 m - 5) \cdot (m^{2} - 5 m - 6)$

Step 1

Expand $(3 m^{3} + m^{2} - 2 m - 5) (m^{2} - 5 m - 6)$ by multiplying each term in the first expression by each term in the second expression.

$3 m^{3} m^{2} + 3 m^{3} (- 5 m) + 3 m^{3} \cdot - 6 + m^{2} m^{2} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $m^{3}$ by $m^{2}$ by adding the exponents.

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Step 2.1.1.1

Move $m^{2}$ .

$3 (m^{2} m^{3}) + 3 m^{3} (- 5 m) + 3 m^{3} \cdot - 6 + m^{2} m^{2} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.1.2

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

$3 m^{2 + 3} + 3 m^{3} (- 5 m) + 3 m^{3} \cdot - 6 + m^{2} m^{2} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.1.3

Add $2$ and $3$ .

$3 m^{5} + 3 m^{3} (- 5 m) + 3 m^{3} \cdot - 6 + m^{2} m^{2} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$3 m^{5} + 3 \cdot - 5 m^{3} m + 3 m^{3} \cdot - 6 + m^{2} m^{2} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.3

Multiply $m^{3}$ by $m$ by adding the exponents.

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Step 2.1.3.1

Move $m$ .

$3 m^{5} + 3 \cdot - 5 (m \cdot m^{3}) + 3 m^{3} \cdot - 6 + m^{2} m^{2} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.3.2

Multiply $m$ by $m^{3}$ .

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Step 2.1.3.2.1

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

$3 m^{5} + 3 \cdot - 5 (m^{1} m^{3}) + 3 m^{3} \cdot - 6 + m^{2} m^{2} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.3.2.2

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

$3 m^{5} + 3 \cdot - 5 m^{1 + 3} + 3 m^{3} \cdot - 6 + m^{2} m^{2} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.3.3

Add $1$ and $3$ .

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

Step 2.1.4

Multiply $3$ by $- 5$ .

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

Step 2.1.5

Multiply $- 6$ by $3$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{2} m^{2} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.6

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

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Step 2.1.6.1

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

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{2 + 2} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.6.2

Add $2$ and $2$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} + m^{2} (- 5 m) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.7

Rewrite using the commutative property of multiplication.

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{2} m + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.8

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

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Step 2.1.8.1

Move $m$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 (m \cdot m^{2}) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.8.2

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

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Step 2.1.8.2.1

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

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 (m^{1} m^{2}) + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.8.2.2

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

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{1 + 2} + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.8.3

Add $1$ and $2$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} + m^{2} \cdot - 6 - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.9

Move $- 6$ to the left of $m^{2}$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 \cdot m^{2} - 2 m \cdot m^{2} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.10

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

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Step 2.1.10.1

Move $m^{2}$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 (m^{2} m) - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.10.2

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

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Step 2.1.10.2.1

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

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 (m^{2} m^{1}) - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.10.2.2

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

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 m^{2 + 1} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.10.3

Add $2$ and $1$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 m^{3} - 2 m (- 5 m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.11

Rewrite using the commutative property of multiplication.

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 m^{3} - 2 \cdot - 5 m \cdot m - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.12

Multiply $m$ by $m$ by adding the exponents.

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Step 2.1.12.1

Move $m$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 m^{3} - 2 \cdot - 5 (m \cdot m) - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.12.2

Multiply $m$ by $m$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 m^{3} - 2 \cdot - 5 m^{2} - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.13

Multiply $- 2$ by $- 5$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 m^{3} + 10 m^{2} - 2 m \cdot - 6 - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.14

Multiply $- 6$ by $- 2$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 m^{3} + 10 m^{2} + 12 m - 5 m^{2} - 5 (- 5 m) - 5 \cdot - 6$

Step 2.1.15

Multiply $- 5$ by $- 5$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 m^{3} + 10 m^{2} + 12 m - 5 m^{2} + 25 m - 5 \cdot - 6$

Step 2.1.16

Multiply $- 5$ by $- 6$ .

$3 m^{5} - 15 m^{4} - 18 m^{3} + m^{4} - 5 m^{3} - 6 m^{2} - 2 m^{3} + 10 m^{2} + 12 m - 5 m^{2} + 25 m + 30$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $- 15 m^{4}$ and $m^{4}$ .

$3 m^{5} - 14 m^{4} - 18 m^{3} - 5 m^{3} - 6 m^{2} - 2 m^{3} + 10 m^{2} + 12 m - 5 m^{2} + 25 m + 30$

Step 2.2.2

Subtract $5 m^{3}$ from $- 18 m^{3}$ .

$3 m^{5} - 14 m^{4} - 23 m^{3} - 6 m^{2} - 2 m^{3} + 10 m^{2} + 12 m - 5 m^{2} + 25 m + 30$

Step 2.2.3

Subtract $2 m^{3}$ from $- 23 m^{3}$ .

$3 m^{5} - 14 m^{4} - 25 m^{3} - 6 m^{2} + 10 m^{2} + 12 m - 5 m^{2} + 25 m + 30$

Step 2.2.4

Add $- 6 m^{2}$ and $10 m^{2}$ .

$3 m^{5} - 14 m^{4} - 25 m^{3} + 4 m^{2} + 12 m - 5 m^{2} + 25 m + 30$

Step 2.2.5

Subtract $5 m^{2}$ from $4 m^{2}$ .

$3 m^{5} - 14 m^{4} - 25 m^{3} - m^{2} + 12 m + 25 m + 30$

Step 2.2.6

Add $12 m$ and $25 m$ .

$3 m^{5} - 14 m^{4} - 25 m^{3} - m^{2} + 37 m + 30$