Basic Math Examples

$120 n^{4} + 116 n^{3} d - 28 n^{2} d^{2}$

Step 1

Factor $4 n^{2}$ out of $120 n^{4} + 116 n^{3} d - 28 n^{2} d^{2}$ .

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Step 1.1

Factor $4 n^{2}$ out of $120 n^{4}$ .

$4 n^{2} (30 n^{2}) + 116 n^{3} d - 28 n^{2} d^{2}$

Step 1.2

Factor $4 n^{2}$ out of $116 n^{3} d$ .

$4 n^{2} (30 n^{2}) + 4 n^{2} (29 n d) - 28 n^{2} d^{2}$

Step 1.3

Factor $4 n^{2}$ out of $- 28 n^{2} d^{2}$ .

$4 n^{2} (30 n^{2}) + 4 n^{2} (29 n d) + 4 n^{2} (- 7 d^{2})$

Step 1.4

Factor $4 n^{2}$ out of $4 n^{2} (30 n^{2}) + 4 n^{2} (29 n d)$ .

$4 n^{2} (30 n^{2} + 29 n d) + 4 n^{2} (- 7 d^{2})$

Step 1.5

Factor $4 n^{2}$ out of $4 n^{2} (30 n^{2} + 29 n d) + 4 n^{2} (- 7 d^{2})$ .

$4 n^{2} (30 n^{2} + 29 n d - 7 d^{2})$

Step 2

Factor.

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Step 2.1

Factor by grouping.

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Step 2.1.1

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 $a \cdot c = - 7 \cdot 30 = - 210$ and whose sum is $b = 29$ .

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Step 2.1.1.1

Reorder terms.

$4 n^{2} (- 7 d^{2} + 30 n^{2} + 29 d n)$

Step 2.1.1.2

Reorder $30 n^{2}$ and $29 d n$ .

$4 n^{2} (- 7 d^{2} + 29 d n + 30 n^{2})$

Step 2.1.1.3

Factor $29$ out of $29 d n$ .

$4 n^{2} (- 7 d^{2} + 29 (d n) + 30 n^{2})$

Step 2.1.1.4

Rewrite $29$ as $- 6$ plus $35$

$4 n^{2} (- 7 d^{2} + (- 6 + 35) (d n) + 30 n^{2})$

Step 2.1.1.5

Apply the distributive property.

$4 n^{2} (- 7 d^{2} - 6 (d n) + 35 (d n) + 30 n^{2})$

Step 2.1.1.6

Remove unnecessary parentheses.

$4 n^{2} (- 7 d^{2} - 6 d n + 35 (d n) + 30 n^{2})$

Step 2.1.1.7

Remove unnecessary parentheses.

$4 n^{2} (- 7 d^{2} - 6 d n + 35 d n + 30 n^{2})$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$4 n^{2} ((- 7 d^{2} - 6 d n) + 35 d n + 30 n^{2})$

Step 2.1.2.2

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

$4 n^{2} (d (- 7 d - 6 n) - 5 n (- 7 d - 6 n))$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $- 7 d - 6 n$ .

$4 n^{2} ((- 7 d - 6 n) (d - 5 n))$

Step 2.2

Remove unnecessary parentheses.

$4 n^{2} (- 7 d - 6 n) (d - 5 n)$

Step 3

Factor.

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Step 3.1

Factor $- 1$ out of $- 7 d - 6 n$ .

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Step 3.1.1

Factor $- 1$ out of $- 7 d$ .

$4 n^{2} (- (7 d) - 6 n) (d - 5 n)$

Step 3.1.2

Factor $- 1$ out of $- 6 n$ .

$4 n^{2} (- (7 d) - (6 n)) (d - 5 n)$

Step 3.1.3

Factor $- 1$ out of $- (7 d) - (6 n)$ .

$4 n^{2} (- (7 d + 6 n)) (d - 5 n)$

Step 3.2

Remove unnecessary parentheses.

$4 n^{2} \cdot - 1 (7 d + 6 n) (d - 5 n)$

Step 4

Factor.

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Step 4.1

Combine exponents.

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Step 4.1.1

Factor out negative.

$- ((4 n^{2} (7 d + 6 n)) (d - 5 n))$

Step 4.1.2

Multiply $4$ by $- 1$ .

$- 4 ((n^{2} (7 d + 6 n)) (d - 5 n))$

Step 4.2

Remove unnecessary parentheses.

$- 4 n^{2} (7 d + 6 n) (d - 5 n)$