Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.1.2

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

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Step 2.1.2.1

Move $x^{2}$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Add $2$ and $2$ .

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

Step 2.1.3

Multiply $3$ by $2$ .

$6 x^{4} + 3 x^{2} x + 3 x^{2} \cdot 3 + x (2 x^{2}) + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.4

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

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Step 2.1.4.1

Move $x$ .

$6 x^{4} + 3 (x \cdot x^{2}) + 3 x^{2} \cdot 3 + x (2 x^{2}) + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.4.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.4.2.1

Raise $x$ to the power of $1$ .

$6 x^{4} + 3 (x^{1} x^{2}) + 3 x^{2} \cdot 3 + x (2 x^{2}) + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.4.2.2

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

$6 x^{4} + 3 x^{1 + 2} + 3 x^{2} \cdot 3 + x (2 x^{2}) + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.4.3

Add $1$ and $2$ .

$6 x^{4} + 3 x^{3} + 3 x^{2} \cdot 3 + x (2 x^{2}) + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.5

Multiply $3$ by $3$ .

$6 x^{4} + 3 x^{3} + 9 x^{2} + x (2 x^{2}) + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$6 x^{4} + 3 x^{3} + 9 x^{2} + 2 x \cdot x^{2} + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.7

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

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Step 2.1.7.1

Move $x^{2}$ .

$6 x^{4} + 3 x^{3} + 9 x^{2} + 2 (x^{2} x) + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.7.2

Multiply $x^{2}$ by $x$ .

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Step 2.1.7.2.1

Raise $x$ to the power of $1$ .

$6 x^{4} + 3 x^{3} + 9 x^{2} + 2 (x^{2} x^{1}) + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.7.2.2

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

$6 x^{4} + 3 x^{3} + 9 x^{2} + 2 x^{2 + 1} + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.7.3

Add $2$ and $1$ .

$6 x^{4} + 3 x^{3} + 9 x^{2} + 2 x^{3} + x \cdot x + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.8

Multiply $x$ by $x$ .

$6 x^{4} + 3 x^{3} + 9 x^{2} + 2 x^{3} + x^{2} + x \cdot 3 - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.9

Move $3$ to the left of $x$ .

$6 x^{4} + 3 x^{3} + 9 x^{2} + 2 x^{3} + x^{2} + 3 \cdot x - 5 (2 x^{2}) - 5 x - 5 \cdot 3$

Step 2.1.10

Multiply $2$ by $- 5$ .

$6 x^{4} + 3 x^{3} + 9 x^{2} + 2 x^{3} + x^{2} + 3 x - 10 x^{2} - 5 x - 5 \cdot 3$

Step 2.1.11

Multiply $- 5$ by $3$ .

$6 x^{4} + 3 x^{3} + 9 x^{2} + 2 x^{3} + x^{2} + 3 x - 10 x^{2} - 5 x - 15$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $3 x^{3}$ and $2 x^{3}$ .

$6 x^{4} + 5 x^{3} + 9 x^{2} + x^{2} + 3 x - 10 x^{2} - 5 x - 15$

Step 2.2.2

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

$6 x^{4} + 5 x^{3} + 10 x^{2} + 3 x - 10 x^{2} - 5 x - 15$

Step 2.2.3

Combine the opposite terms in $6 x^{4} + 5 x^{3} + 10 x^{2} + 3 x - 10 x^{2} - 5 x - 15$ .

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Step 2.2.3.1

Subtract $10 x^{2}$ from $10 x^{2}$ .

$6 x^{4} + 5 x^{3} + 3 x + 0 - 5 x - 15$

Step 2.2.3.2

Add $6 x^{4} + 5 x^{3} + 3 x$ and $0$ .

$6 x^{4} + 5 x^{3} + 3 x - 5 x - 15$

Step 2.2.4

Subtract $5 x$ from $3 x$ .

$6 x^{4} + 5 x^{3} - 2 x - 15$