Pre-Algebra Examples

$\frac{{(m x^{2} + 1)}^{2} + {(m + x)}^{2} + m x}{x + 1}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{(m x^{2} + 1) (m x^{2} + 1) + {(m + x)}^{2} + m x}{x + 1}$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$\frac{m x^{2} (m x^{2} + 1) + 1 (m x^{2} + 1) + {(m + x)}^{2} + m x}{x + 1}$

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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

Multiply $m$ by $m$ by adding the exponents.

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Step 1.3.1.1.1

Move $m$ .

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

Step 1.3.1.1.2

Multiply $m$ by $m$ .

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

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 $x^{2}$ .

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

Step 1.3.1.2.2

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

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

Step 1.3.1.2.3

Add $2$ and $2$ .

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

Step 1.3.1.3

Multiply $m$ by $1$ .

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

Step 1.3.1.4

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

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

Step 1.3.1.5

Multiply $1$ by $1$ .

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

Step 1.3.2

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

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

Step 1.4

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

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

Step 1.5

Expand $(m + x) (m + x)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

$\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m \cdot m + m x + x m + x \cdot x + m x}{x + 1}$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $m$ by $m$ .

$\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m^{2} + m x + x m + x \cdot x + m x}{x + 1}$

Step 1.6.1.2

Multiply $x$ by $x$ .

$\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m^{2} + m x + x m + x^{2} + m x}{x + 1}$

Step 1.6.2

Add $m x$ and $x m$ .

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Step 1.6.2.1

Reorder $x$ and $m$ .

$\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m^{2} + m x + m x + x^{2} + m x}{x + 1}$

Step 1.6.2.2

Add $m x$ and $m x$ .

$\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m^{2} + 2 m x + x^{2} + m x}{x + 1}$

Step 1.7

Add $2 m x$ and $m x$ .

$\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m^{2} + x^{2} + 3 m x}{x + 1}$

Step 2

Split the fraction $\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m^{2} + x^{2} + 3 m x}{x + 1}$ into two fractions.

$\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m^{2} + x^{2}}{x + 1} + \frac{3 m x}{x + 1}$

Step 3

Split the fraction $\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m^{2} + x^{2}}{x + 1}$ into two fractions.

$\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m^{2}}{x + 1} + \frac{x^{2}}{x + 1} + \frac{3 m x}{x + 1}$

Step 4

Split the fraction $\frac{m^{2} x^{4} + 2 m x^{2} + 1 + m^{2}}{x + 1}$ into two fractions.

$\frac{m^{2} x^{4} + 2 m x^{2} + 1}{x + 1} + \frac{m^{2}}{x + 1} + \frac{x^{2}}{x + 1} + \frac{3 m x}{x + 1}$

Step 5

Split the fraction $\frac{m^{2} x^{4} + 2 m x^{2} + 1}{x + 1}$ into two fractions.

$\frac{m^{2} x^{4} + 2 m x^{2}}{x + 1} + \frac{1}{x + 1} + \frac{m^{2}}{x + 1} + \frac{x^{2}}{x + 1} + \frac{3 m x}{x + 1}$

Step 6

Split the fraction $\frac{m^{2} x^{4} + 2 m x^{2}}{x + 1}$ into two fractions.

$\frac{m^{2} x^{4}}{x + 1} + \frac{2 m x^{2}}{x + 1} + \frac{1}{x + 1} + \frac{m^{2}}{x + 1} + \frac{x^{2}}{x + 1} + \frac{3 m x}{x + 1}$