Precalculus Examples

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

Step 1

Simplify terms.

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Step 1.1

Remove parentheses.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Multiply $- 1$ by $5$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $x \cdot x + x \cdot - 5 + x (- x^{2}) + 5 x + 5 \cdot - 5 + 5 (- x^{2}) + x^{2} x + x^{2} \cdot - 5 + x^{2} (- x^{2})$ .

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Step 3.1.1

Reorder the factors in the terms $x \cdot - 5$ and $5 x$ .

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

Step 3.1.2

Add $- 5 x$ and $5 x$ .

$x \cdot x + x (- x^{2}) + 0 + 5 \cdot - 5 + 5 (- x^{2}) + x^{2} x + x^{2} \cdot - 5 + x^{2} (- x^{2})$

Step 3.1.3

Add $x \cdot x + x (- x^{2})$ and $0$ .

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

Step 3.1.4

Reorder the factors in the terms $x (- x^{2})$ and $x^{2} x$ .

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

Step 3.1.5

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

$x \cdot x + 5 \cdot - 5 + 5 (- x^{2}) + 0 + x^{2} \cdot - 5 + x^{2} (- x^{2})$

Step 3.1.6

Add $x \cdot x + 5 \cdot - 5 + 5 (- x^{2})$ and $0$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x$ by $x$ .

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

Step 3.2.2

Multiply $5$ by $- 5$ .

$x^{2} - 25 + 5 (- x^{2}) + x^{2} \cdot - 5 + x^{2} (- x^{2})$

Step 3.2.3

Multiply $- 1$ by $5$ .

$x^{2} - 25 - 5 x^{2} + x^{2} \cdot - 5 + x^{2} (- x^{2})$

Step 3.2.4

Move $- 5$ to the left of $x^{2}$ .

$x^{2} - 25 - 5 x^{2} - 5 \cdot x^{2} + x^{2} (- x^{2})$

Step 3.2.5

Rewrite using the commutative property of multiplication.

$x^{2} - 25 - 5 x^{2} - 5 x^{2} - x^{2} x^{2}$

Step 3.2.6

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

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Step 3.2.6.1

Move $x^{2}$ .

$x^{2} - 25 - 5 x^{2} - 5 x^{2} - (x^{2} x^{2})$

Step 3.2.6.2

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

$x^{2} - 25 - 5 x^{2} - 5 x^{2} - x^{2 + 2}$

Step 3.2.6.3

Add $2$ and $2$ .

$x^{2} - 25 - 5 x^{2} - 5 x^{2} - x^{4}$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

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

$- 4 x^{2} - 25 - 5 x^{2} - x^{4}$

Step 3.3.2

Subtract $5 x^{2}$ from $- 4 x^{2}$ .

$- 9 x^{2} - 25 - x^{4}$

Step 3.3.3

Simplify the expression.

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Step 3.3.3.1

Move $- 25$ .

$- 9 x^{2} - x^{4} - 25$

Step 3.3.3.2

Reorder $- 9 x^{2}$ and $- x^{4}$ .

$- x^{4} - 9 x^{2} - 25$