Basic Math Examples

$- 20 p {(3 p - 2)}^{2} - 20 (3 p - 2) p - 5 p$

Step 1

Factor $- 5 p$ out of $- 20 p {(3 p - 2)}^{2} - 20 (3 p - 2) p - 5 p$ .

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Step 1.1

Move $3 p - 2$ .

$- 20 p {(3 p - 2)}^{2} - 20 p (3 p - 2) - 5 p$

Step 1.2

Factor $- 5 p$ out of $- 20 p {(3 p - 2)}^{2}$ .

$- 5 p (4 {(3 p - 2)}^{2}) - 20 p (3 p - 2) - 5 p$

Step 1.3

Factor $- 5 p$ out of $- 20 p (3 p - 2)$ .

$- 5 p (4 {(3 p - 2)}^{2}) - 5 p (4 (3 p - 2)) - 5 p$

Step 1.4

Factor $- 5 p$ out of $- 5 p$ .

$- 5 p (4 {(3 p - 2)}^{2}) - 5 p (4 (3 p - 2)) - 5 p (1)$

Step 1.5

Factor $- 5 p$ out of $- 5 p (4 {(3 p - 2)}^{2}) - 5 p (4 (3 p - 2))$ .

$- 5 p (4 {(3 p - 2)}^{2} + 4 (3 p - 2)) - 5 p (1)$

Step 1.6

Factor $- 5 p$ out of $- 5 p (4 {(3 p - 2)}^{2} + 4 (3 p - 2)) - 5 p (1)$ .

$- 5 p (4 {(3 p - 2)}^{2} + 4 (3 p - 2) + 1)$

Step 2

Let $u = 3 p - 2$ . Substitute $u$ for all occurrences of $3 p - 2$ .

$- 5 p (4 u^{2} + 4 u + 1)$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $4 u^{2}$ as ${(2 u)}^{2}$ .

$- 5 p ({(2 u)}^{2} + 4 u + 1)$

Step 3.2

Rewrite $1$ as $1^{2}$ .

$- 5 p ({(2 u)}^{2} + 4 u + 1^{2})$

Step 3.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot (2 u) \cdot 1$

Step 3.4

Rewrite the polynomial.

$- 5 p ({(2 u)}^{2} + 2 \cdot (2 u) \cdot 1 + 1^{2})$

Step 3.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 u$ and $b = 1$ .

$- 5 p {(2 u + 1)}^{2}$

Step 4

Replace all occurrences of $u$ with $3 p - 2$ .

$- 5 p {(2 (3 p - 2) + 1)}^{2}$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Multiply $3$ by $2$ .

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

Step 5.1.3

Multiply $2$ by $- 2$ .

$- 5 p {(6 p - 4 + 1)}^{2}$

Step 5.2

Add $- 4$ and $1$ .

$- 5 p {(6 p - 3)}^{2}$

Step 6

Factor $3$ out of $6 p - 3$ .

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Step 6.1

Factor $3$ out of $6 p$ .

$- 5 p {(3 (2 p) - 3)}^{2}$

Step 6.2

Factor $3$ out of $- 3$ .

$- 5 p {(3 (2 p) + 3 (- 1))}^{2}$

Step 6.3

Factor $3$ out of $3 (2 p) + 3 (- 1)$ .

$- 5 p {(3 (2 p - 1))}^{2}$

Step 7

Factor.

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Step 7.1

Apply the product rule to $3 (2 p - 1)$ .

$- 5 p (3^{2} {(2 p - 1)}^{2})$

Step 7.2

Remove unnecessary parentheses.

$- 5 p \cdot 3^{2} {(2 p - 1)}^{2}$