Algebra Examples

$(4 x^{2} + 3 x + 7) (- 2 x^{2} + x + 2)$

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 $(4 x^{2} + 3 x + 7) (- 2 x^{2} + x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$4 x^{2} (- 2 x^{2}) + 4 x^{2} x + 4 x^{2} \cdot 2 + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$4 \cdot - 2 x^{2} x^{2} + 4 x^{2} x + 4 x^{2} \cdot 2 + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.2

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

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Step 3.1.2.1

Move $x^{2}$ .

$4 \cdot - 2 (x^{2} x^{2}) + 4 x^{2} x + 4 x^{2} \cdot 2 + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.2.2

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

$4 \cdot - 2 x^{2 + 2} + 4 x^{2} x + 4 x^{2} \cdot 2 + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.2.3

Add $2$ and $2$ .

$4 \cdot - 2 x^{4} + 4 x^{2} x + 4 x^{2} \cdot 2 + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.3

Multiply $4$ by $- 2$ .

$- 8 x^{4} + 4 x^{2} x + 4 x^{2} \cdot 2 + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.4

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

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Step 3.1.4.1

Move $x$ .

$- 8 x^{4} + 4 (x \cdot x^{2}) + 4 x^{2} \cdot 2 + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.4.2

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

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Step 3.1.4.2.1

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

$- 8 x^{4} + 4 (x^{1} x^{2}) + 4 x^{2} \cdot 2 + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.4.2.2

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

$- 8 x^{4} + 4 x^{1 + 2} + 4 x^{2} \cdot 2 + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.4.3

Add $1$ and $2$ .

$- 8 x^{4} + 4 x^{3} + 4 x^{2} \cdot 2 + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.5

Multiply $2$ by $4$ .

$- 8 x^{4} + 4 x^{3} + 8 x^{2} + 3 x (- 2 x^{2}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.6

Rewrite using the commutative property of multiplication.

$- 8 x^{4} + 4 x^{3} + 8 x^{2} + 3 \cdot - 2 x \cdot x^{2} + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.7

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

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Step 3.1.7.1

Move $x^{2}$ .

$- 8 x^{4} + 4 x^{3} + 8 x^{2} + 3 \cdot - 2 (x^{2} x) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.7.2

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

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Step 3.1.7.2.1

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

$- 8 x^{4} + 4 x^{3} + 8 x^{2} + 3 \cdot - 2 (x^{2} x^{1}) + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.7.2.2

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

$- 8 x^{4} + 4 x^{3} + 8 x^{2} + 3 \cdot - 2 x^{2 + 1} + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.7.3

Add $2$ and $1$ .

$- 8 x^{4} + 4 x^{3} + 8 x^{2} + 3 \cdot - 2 x^{3} + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.8

Multiply $3$ by $- 2$ .

$- 8 x^{4} + 4 x^{3} + 8 x^{2} - 6 x^{3} + 3 x \cdot x + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.9

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

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Step 3.1.9.1

Move $x$ .

$- 8 x^{4} + 4 x^{3} + 8 x^{2} - 6 x^{3} + 3 (x \cdot x) + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.9.2

Multiply $x$ by $x$ .

$- 8 x^{4} + 4 x^{3} + 8 x^{2} - 6 x^{3} + 3 x^{2} + 3 x \cdot 2 + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.10

Multiply $2$ by $3$ .

$- 8 x^{4} + 4 x^{3} + 8 x^{2} - 6 x^{3} + 3 x^{2} + 6 x + 7 (- 2 x^{2}) + 7 x + 7 \cdot 2$

Step 3.1.11

Multiply $- 2$ by $7$ .

$- 8 x^{4} + 4 x^{3} + 8 x^{2} - 6 x^{3} + 3 x^{2} + 6 x - 14 x^{2} + 7 x + 7 \cdot 2$

Step 3.1.12

Multiply $7$ by $2$ .

$- 8 x^{4} + 4 x^{3} + 8 x^{2} - 6 x^{3} + 3 x^{2} + 6 x - 14 x^{2} + 7 x + 14$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Subtract $6 x^{3}$ from $4 x^{3}$ .

$- 8 x^{4} - 2 x^{3} + 8 x^{2} + 3 x^{2} + 6 x - 14 x^{2} + 7 x + 14$

Step 3.2.2

Add $8 x^{2}$ and $3 x^{2}$ .

$- 8 x^{4} - 2 x^{3} + 11 x^{2} + 6 x - 14 x^{2} + 7 x + 14$

Step 3.2.3

Subtract $14 x^{2}$ from $11 x^{2}$ .

$- 8 x^{4} - 2 x^{3} - 3 x^{2} + 6 x + 7 x + 14$

Step 3.2.4

Add $6 x$ and $7 x$ .

$- 8 x^{4} - 2 x^{3} - 3 x^{2} + 13 x + 14$