Algebra Examples

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

Step 1

Expand $(- 2 x^{2} + x - 2) (x^{2} - 5 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

Move $x^{2}$ .

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

Step 2.1.1.2

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

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

Step 2.1.1.3

Add $2$ and $2$ .

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

Step 2.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.1.3

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

Tap for more steps...

Step 2.1.3.1

Move $x$ .

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

Step 2.1.3.2

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

Tap for more steps...

Step 2.1.3.2.1

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

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

Step 2.1.3.2.2

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

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

Step 2.1.3.3

Add $1$ and $2$ .

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

Step 2.1.4

Multiply $- 2$ by $- 5$ .

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

Step 2.1.5

Multiply $4$ by $- 2$ .

$- 2 x^{4} + 10 x^{3} - 8 x^{2} + x \cdot x^{2} + x (- 5 x) + x \cdot 4 - 2 x^{2} - 2 (- 5 x) - 2 \cdot 4$

Step 2.1.6

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

Tap for more steps...

Step 2.1.6.1

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

Tap for more steps...

Step 2.1.6.1.1

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

$- 2 x^{4} + 10 x^{3} - 8 x^{2} + x^{1} x^{2} + x (- 5 x) + x \cdot 4 - 2 x^{2} - 2 (- 5 x) - 2 \cdot 4$

Step 2.1.6.1.2

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

$- 2 x^{4} + 10 x^{3} - 8 x^{2} + x^{1 + 2} + x (- 5 x) + x \cdot 4 - 2 x^{2} - 2 (- 5 x) - 2 \cdot 4$

Step 2.1.6.2

Add $1$ and $2$ .

$- 2 x^{4} + 10 x^{3} - 8 x^{2} + x^{3} + x (- 5 x) + x \cdot 4 - 2 x^{2} - 2 (- 5 x) - 2 \cdot 4$

Step 2.1.7

Rewrite using the commutative property of multiplication.

$- 2 x^{4} + 10 x^{3} - 8 x^{2} + x^{3} - 5 x \cdot x + x \cdot 4 - 2 x^{2} - 2 (- 5 x) - 2 \cdot 4$

Step 2.1.8

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

Tap for more steps...

Step 2.1.8.1

Move $x$ .

$- 2 x^{4} + 10 x^{3} - 8 x^{2} + x^{3} - 5 (x \cdot x) + x \cdot 4 - 2 x^{2} - 2 (- 5 x) - 2 \cdot 4$

Step 2.1.8.2

Multiply $x$ by $x$ .

$- 2 x^{4} + 10 x^{3} - 8 x^{2} + x^{3} - 5 x^{2} + x \cdot 4 - 2 x^{2} - 2 (- 5 x) - 2 \cdot 4$

Step 2.1.9

Move $4$ to the left of $x$ .

$- 2 x^{4} + 10 x^{3} - 8 x^{2} + x^{3} - 5 x^{2} + 4 \cdot x - 2 x^{2} - 2 (- 5 x) - 2 \cdot 4$

Step 2.1.10

Multiply $- 5$ by $- 2$ .

$- 2 x^{4} + 10 x^{3} - 8 x^{2} + x^{3} - 5 x^{2} + 4 x - 2 x^{2} + 10 x - 2 \cdot 4$

Step 2.1.11

Multiply $- 2$ by $4$ .

$- 2 x^{4} + 10 x^{3} - 8 x^{2} + x^{3} - 5 x^{2} + 4 x - 2 x^{2} + 10 x - 8$

Step 2.2

Simplify by adding terms.

Tap for more steps...

Step 2.2.1

Add $10 x^{3}$ and $x^{3}$ .

$- 2 x^{4} + 11 x^{3} - 8 x^{2} - 5 x^{2} + 4 x - 2 x^{2} + 10 x - 8$

Step 2.2.2

Subtract $5 x^{2}$ from $- 8 x^{2}$ .

$- 2 x^{4} + 11 x^{3} - 13 x^{2} + 4 x - 2 x^{2} + 10 x - 8$

Step 2.2.3

Subtract $2 x^{2}$ from $- 13 x^{2}$ .

$- 2 x^{4} + 11 x^{3} - 15 x^{2} + 4 x + 10 x - 8$

Step 2.2.4

Add $4 x$ and $10 x$ .

$- 2 x^{4} + 11 x^{3} - 15 x^{2} + 14 x - 8$