Basic Math Examples

$2 n^{2} - 3 (n m) - 3 n^{2} + 6 (n m) + 4 m^{2}$

Step 1

Remove parentheses.

$2 n^{2} - 3 n m - 3 n^{2} + 6 (n m) + 4 m^{2}$

Step 2

Remove parentheses.

$2 n^{2} - 3 n m - 3 n^{2} + 6 n m + 4 m^{2}$

Step 3

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

$- n^{2} - 3 n m + 6 n m + 4 m^{2}$

Step 4

Add $- 3 n m$ and $6 n m$ .

$- n^{2} + 3 n m + 4 m^{2}$

Step 5

Factor $- 1$ out of $- n^{2} + 3 n m + 4 m^{2}$ .

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Step 5.1

Reorder the expression.

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Step 5.1.1

Move $n$ .

$- n^{2} + 3 m n + 4 m^{2}$

Step 5.1.2

Move $- n^{2}$ .

$3 m n + 4 m^{2} - n^{2}$

Step 5.1.3

Reorder $3 m n$ and $4 m^{2}$ .

$4 m^{2} + 3 m n - n^{2}$

Step 5.2

Factor $- 1$ out of $4 m^{2}$ .

$- (- 4 m^{2}) + 3 m n - n^{2}$

Step 5.3

Factor $- 1$ out of $3 m n$ .

$- (- 4 m^{2}) - (- 3 m n) - n^{2}$

Step 5.4

Factor $- 1$ out of $- n^{2}$ .

$- (- 4 m^{2}) - (- 3 m n) - (n^{2})$

Step 5.5

Factor $- 1$ out of $- (- 4 m^{2}) - (- 3 m n)$ .

$- (- 4 m^{2} - 3 m n) - (n^{2})$

Step 5.6

Factor $- 1$ out of $- (- 4 m^{2} - 3 m n) - (n^{2})$ .

$- (- 4 m^{2} - 3 m n + n^{2})$

Step 6

Factor.

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Step 6.1

Factor by grouping.

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Step 6.1.1

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 $a \cdot c = - 4 \cdot 1 = - 4$ and whose sum is $b = - 3$ .

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Step 6.1.1.1

Reorder terms.

$- (- 4 m^{2} + n^{2} - 3 m n)$

Step 6.1.1.2

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

$- (- 4 m^{2} - 3 m n + n^{2})$

Step 6.1.1.3

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

$- (- 4 m^{2} - 3 (m n) + n^{2})$

Step 6.1.1.4

Rewrite $- 3$ as $1$ plus $- 4$

$- (- 4 m^{2} + (1 - 4) (m n) + n^{2})$

Step 6.1.1.5

Apply the distributive property.

$- (- 4 m^{2} + 1 (m n) - 4 (m n) + n^{2})$

Step 6.1.1.6

Multiply $m n$ by $1$ .

$- (- 4 m^{2} + m n - 4 (m n) + n^{2})$

Step 6.1.1.7

Remove unnecessary parentheses.

$- (- 4 m^{2} + m n - 4 m n + n^{2})$

Step 6.1.2

Factor out the greatest common factor from each group.

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Step 6.1.2.1

Group the first two terms and the last two terms.

$- ((- 4 m^{2} + m n) - 4 m n + n^{2})$

Step 6.1.2.2

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

$- (m (- 4 m + n) + n (- 4 m + n))$

Step 6.1.3

Factor the polynomial by factoring out the greatest common factor, $- 4 m + n$ .

$- ((- 4 m + n) (m + n))$

Step 6.2

Remove unnecessary parentheses.

$- (- 4 m + n) (m + n)$

Step 7

Factor.

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Step 7.1

Factor $- 1$ out of $- 4 m + n$ .

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Step 7.1.1

Factor $- 1$ out of $- 4 m$ .

$- (- (4 m) + n) (m + n)$

Step 7.1.2

Factor $- 1$ out of $n$ .

$- (- (4 m) - 1 (- n)) (m + n)$

Step 7.1.3

Factor $- 1$ out of $- (4 m) - 1 (- n)$ .

$- - (4 m - n) (m + n)$

Step 7.2

Remove unnecessary parentheses.

$- - 1 (4 m - n) (m + n)$

Step 8

Combine exponents.

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Step 8.1

Factor out negative.

$- ((- (4 m - n)) (m + n))$

Step 8.2

Multiply $- 1$ by $- 1$ .

$1 ((4 m - n) (m + n))$

Step 8.3

Multiply $4 m - n$ by $1$ .

$(4 m - n) (m + n)$