Algebra Examples

$2 {(2 x^{2} + 1)}^{2} + 1$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(2 x^{2} + 1)}^{2}$ as $(2 x^{2} + 1) (2 x^{2} + 1)$ .

$2 ((2 x^{2} + 1) (2 x^{2} + 1)) + 1$

Step 1.2

Expand $(2 x^{2} + 1) (2 x^{2} + 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$2 (2 x^{2} (2 x^{2} + 1) + 1 (2 x^{2} + 1)) + 1$

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x^{2}$ .

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

Step 1.3.1.2.2

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

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

Step 1.3.1.2.3

Add $2$ and $2$ .

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

Step 1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.3.1.4

Multiply $2$ by $1$ .

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

Step 1.3.1.5

Multiply $2 x^{2}$ by $1$ .

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

Step 1.3.1.6

Multiply $1$ by $1$ .

$2 (4 x^{4} + 2 x^{2} + 2 x^{2} + 1) + 1$

Step 1.3.2

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

$2 (4 x^{4} + 4 x^{2} + 1) + 1$

Step 1.4

Apply the distributive property.

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

Step 1.5

Simplify.

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Step 1.5.1

Multiply $4$ by $2$ .

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

Step 1.5.2

Multiply $4$ by $2$ .

$8 x^{4} + 8 x^{2} + 2 \cdot 1 + 1$

Step 1.5.3

Multiply $2$ by $1$ .

$8 x^{4} + 8 x^{2} + 2 + 1$

Step 2

Add $2$ and $1$ .

$8 x^{4} + 8 x^{2} + 3$