Pre-Algebra Examples

$(2 t^{2} - t - 4) (2 t^{2} + 2 t - 1)$

Step 1

Expand $(2 t^{2} - t - 4) (2 t^{2} + 2 t - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.1.2

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

Tap for more steps...

Step 2.1.2.1

Move $t^{2}$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Add $2$ and $2$ .

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

Step 2.1.3

Multiply $2$ by $2$ .

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

Step 2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.5

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

Tap for more steps...

Step 2.1.5.1

Move $t$ .

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

Step 2.1.5.2

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

Tap for more steps...

Step 2.1.5.2.1

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

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

Step 2.1.5.2.2

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

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

Step 2.1.5.3

Add $1$ and $2$ .

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

Step 2.1.6

Multiply $2$ by $2$ .

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

Step 2.1.7

Multiply $- 1$ by $2$ .

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

Step 2.1.8

Rewrite using the commutative property of multiplication.

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

Step 2.1.9

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

Tap for more steps...

Step 2.1.9.1

Move $t^{2}$ .

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

Step 2.1.9.2

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

Tap for more steps...

Step 2.1.9.2.1

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

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

Step 2.1.9.2.2

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

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

Step 2.1.9.3

Add $2$ and $1$ .

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

Step 2.1.10

Multiply $- 1$ by $2$ .

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

Step 2.1.11

Rewrite using the commutative property of multiplication.

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

Step 2.1.12

Multiply $t$ by $t$ by adding the exponents.

Tap for more steps...

Step 2.1.12.1

Move $t$ .

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

Step 2.1.12.2

Multiply $t$ by $t$ .

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

Step 2.1.13

Multiply $- 1$ by $2$ .

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

Step 2.1.14

Multiply $- t \cdot - 1$ .

Tap for more steps...

Step 2.1.14.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.14.2

Multiply $t$ by $1$ .

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

Step 2.1.15

Multiply $2$ by $- 4$ .

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

Step 2.1.16

Multiply $2$ by $- 4$ .

$4 t^{4} + 4 t^{3} - 2 t^{2} - 2 t^{3} - 2 t^{2} + t - 8 t^{2} - 8 t - 4 \cdot - 1$

Step 2.1.17

Multiply $- 4$ by $- 1$ .

$4 t^{4} + 4 t^{3} - 2 t^{2} - 2 t^{3} - 2 t^{2} + t - 8 t^{2} - 8 t + 4$

Step 2.2

Simplify by adding terms.

Tap for more steps...

Step 2.2.1

Subtract $2 t^{3}$ from $4 t^{3}$ .

$4 t^{4} + 2 t^{3} - 2 t^{2} - 2 t^{2} + t - 8 t^{2} - 8 t + 4$

Step 2.2.2

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

$4 t^{4} + 2 t^{3} - 4 t^{2} + t - 8 t^{2} - 8 t + 4$

Step 2.2.3

Subtract $8 t^{2}$ from $- 4 t^{2}$ .

$4 t^{4} + 2 t^{3} - 12 t^{2} + t - 8 t + 4$

Step 2.2.4

Subtract $8 t$ from $t$ .

$4 t^{4} + 2 t^{3} - 12 t^{2} - 7 t + 4$