Algebra Examples

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

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

Expand $(- 2 x^{2} + 3 x - 1) (2 x^{2} + 3 x + 5)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $- 2 x^{2} (2 x^{2}) - 2 x^{2} (3 x) - 2 x^{2} \cdot 5 + 3 x (2 x^{2}) + 3 x (3 x) + 3 x \cdot 5 - 1 (2 x^{2}) - 1 (3 x) - 1 \cdot 5$ .

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Step 3.1.1

Reorder the factors in the terms $- 2 x^{2} (3 x)$ and $3 x (2 x^{2})$ .

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

Step 3.1.2

Add $- 2 \cdot 3 x^{2} x$ and $2 \cdot 3 x^{2} x$ .

$- 2 x^{2} (2 x^{2}) - 2 x^{2} \cdot 5 + 0 + 3 x (3 x) + 3 x \cdot 5 - 1 (2 x^{2}) - 1 (3 x) - 1 \cdot 5$

Step 3.1.3

Add $- 2 x^{2} (2 x^{2}) - 2 x^{2} \cdot 5$ and $0$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2

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

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Step 3.2.2.1

Move $x^{2}$ .

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

Step 3.2.2.2

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

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

Step 3.2.2.3

Add $2$ and $2$ .

$- 2 \cdot 2 x^{4} - 2 x^{2} \cdot 5 + 3 x (3 x) + 3 x \cdot 5 - 1 (2 x^{2}) - 1 (3 x) - 1 \cdot 5$

Step 3.2.3

Multiply $- 2$ by $2$ .

$- 4 x^{4} - 2 x^{2} \cdot 5 + 3 x (3 x) + 3 x \cdot 5 - 1 (2 x^{2}) - 1 (3 x) - 1 \cdot 5$

Step 3.2.4

Multiply $5$ by $- 2$ .

$- 4 x^{4} - 10 x^{2} + 3 x (3 x) + 3 x \cdot 5 - 1 (2 x^{2}) - 1 (3 x) - 1 \cdot 5$

Step 3.2.5

Rewrite using the commutative property of multiplication.

$- 4 x^{4} - 10 x^{2} + 3 \cdot 3 x \cdot x + 3 x \cdot 5 - 1 (2 x^{2}) - 1 (3 x) - 1 \cdot 5$

Step 3.2.6

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.6.1

Move $x$ .

$- 4 x^{4} - 10 x^{2} + 3 \cdot 3 (x \cdot x) + 3 x \cdot 5 - 1 (2 x^{2}) - 1 (3 x) - 1 \cdot 5$

Step 3.2.6.2

Multiply $x$ by $x$ .

$- 4 x^{4} - 10 x^{2} + 3 \cdot 3 x^{2} + 3 x \cdot 5 - 1 (2 x^{2}) - 1 (3 x) - 1 \cdot 5$

Step 3.2.7

Multiply $3$ by $3$ .

$- 4 x^{4} - 10 x^{2} + 9 x^{2} + 3 x \cdot 5 - 1 (2 x^{2}) - 1 (3 x) - 1 \cdot 5$

Step 3.2.8

Multiply $5$ by $3$ .

$- 4 x^{4} - 10 x^{2} + 9 x^{2} + 15 x - 1 (2 x^{2}) - 1 (3 x) - 1 \cdot 5$

Step 3.2.9

Multiply $2$ by $- 1$ .

$- 4 x^{4} - 10 x^{2} + 9 x^{2} + 15 x - 2 x^{2} - 1 (3 x) - 1 \cdot 5$

Step 3.2.10

Multiply $3$ by $- 1$ .

$- 4 x^{4} - 10 x^{2} + 9 x^{2} + 15 x - 2 x^{2} - 3 x - 1 \cdot 5$

Step 3.2.11

Multiply $- 1$ by $5$ .

$- 4 x^{4} - 10 x^{2} + 9 x^{2} + 15 x - 2 x^{2} - 3 x - 5$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Add $- 10 x^{2}$ and $9 x^{2}$ .

$- 4 x^{4} - x^{2} + 15 x - 2 x^{2} - 3 x - 5$

Step 3.3.2

Subtract $2 x^{2}$ from $- x^{2}$ .

$- 4 x^{4} - 3 x^{2} + 15 x - 3 x - 5$

Step 3.3.3

Subtract $3 x$ from $15 x$ .

$- 4 x^{4} - 3 x^{2} + 12 x - 5$