Pre-Algebra Examples

$3 m (m + n) - (m - n) (m + n)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$3 m \cdot m + 3 m n - (m - n) (m + n)$

Step 1.2

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

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Step 1.2.1

Move $m$ .

$3 (m \cdot m) + 3 m n - (m - n) (m + n)$

Step 1.2.2

Multiply $m$ by $m$ .

$3 m^{2} + 3 m n - (m - n) (m + n)$

Step 1.3

Apply the distributive property.

$3 m^{2} + 3 m n + (- m - - n) (m + n)$

Step 1.4

Multiply $- - n$ .

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Step 1.4.1

Multiply $- 1$ by $- 1$ .

$3 m^{2} + 3 m n + (- m + 1 n) (m + n)$

Step 1.4.2

Multiply $n$ by $1$ .

$3 m^{2} + 3 m n + (- m + n) (m + n)$

Step 1.5

Expand $(- m + n) (m + n)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

$3 m^{2} + 3 m n - m (m + n) + n (m + n)$

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

$3 m^{2} + 3 m n - m \cdot m - m n + n m + n \cdot n$

Step 1.6

Combine the opposite terms in $- m \cdot m - m n + n m + n \cdot n$ .

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Step 1.6.1

Reorder the factors in the terms $- m n$ and $n m$ .

$3 m^{2} + 3 m n - m \cdot m - m n + m n + n \cdot n$

Step 1.6.2

Add $- m n$ and $m n$ .

$3 m^{2} + 3 m n - m \cdot m + 0 + n \cdot n$

Step 1.6.3

Add $- m \cdot m$ and $0$ .

$3 m^{2} + 3 m n - m \cdot m + n \cdot n$

Step 1.7

Simplify each term.

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Step 1.7.1

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

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Step 1.7.1.1

Move $m$ .

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

Step 1.7.1.2

Multiply $m$ by $m$ .

$3 m^{2} + 3 m n - m^{2} + n \cdot n$

Step 1.7.2

Multiply $n$ by $n$ .

$3 m^{2} + 3 m n - m^{2} + n^{2}$

Step 2

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

$2 m^{2} + 3 m n + n^{2}$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = 3$ .

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Step 3.1.1

Reorder terms.

$2 m^{2} + n^{2} + 3 m n$

Step 3.1.2

Reorder $n^{2}$ and $3 m n$ .

$2 m^{2} + 3 m n + n^{2}$

Step 3.1.3

Factor $3$ out of $3 m n$ .

$2 m^{2} + 3 (m n) + n^{2}$

Step 3.1.4

Rewrite $3$ as $1$ plus $2$

$2 m^{2} + (1 + 2) (m n) + n^{2}$

Step 3.1.5

Apply the distributive property.

$2 m^{2} + 1 (m n) + 2 (m n) + n^{2}$

Step 3.1.6

Multiply $m n$ by $1$ .

$2 m^{2} + m n + 2 (m n) + n^{2}$

Step 3.1.7

Remove unnecessary parentheses.

$2 m^{2} + m n + 2 m n + n^{2}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(2 m^{2} + m n) + 2 m n + n^{2}$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$m (2 m + n) + n (2 m + n)$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 m + n$ .

$(2 m + n) (m + n)$