Simplify (4x^4+4)/(x^2+1)-8/(x^2+1)

$\frac{4 x^{4} + 4}{x^{2} + 1} - \frac{8}{x^{2} + 1}$

Step 1

Combine fractions.

Combine the numerators over the common denominator.

$\frac{4 x^{4} + 4 - 8}{x^{2} + 1}$

Subtract $8$ from $4$ .

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

Step 2

Simplify the numerator.

Factor $4$ out of $4 x^{4} - 4$ .

Factor $4$ out of $4 x^{4}$ .

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

Factor $4$ out of $- 4$ .

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

Factor $4$ out of $4 (x^{4}) + 4 (- 1)$ .

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

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Rewrite $1$ as $1^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 1$ .

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

Simplify.

Rewrite $1$ as $1^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

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

Step 3

Simplify terms.

Cancel the common factor of $x^{2} + 1$ .

Cancel the common factor.

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

Divide $(4 (x + 1)) (x - 1)$ by $1$ .

$(4 (x + 1)) (x - 1)$

Apply the distributive property.

$(4 x + 4 \cdot 1) (x - 1)$

Multiply $4$ by $1$ .

$(4 x + 4) (x - 1)$

Step 4

Expand $(4 x + 4) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$4 x (x - 1) + 4 (x - 1)$

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 1 + 4 (x - 1)$

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 5

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$4 (x \cdot x) + 4 x \cdot - 1 + 4 x + 4 \cdot - 1$

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $4$ .

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

Multiply $4$ by $- 1$ .

$4 x^{2} - 4 x + 4 x - 4$

Add $- 4 x$ and $4 x$ .

$4 x^{2} + 0 - 4$

Add $4 x^{2}$ and $0$ .

$4 x^{2} - 4$

Simplify (4x^4+4)/(x^2+1)-8/(x^2+1)

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