Pre-Algebra Examples

$\frac{x + 3}{4 x + 4} \div \frac{x^{2} \cdot (2 x) - 3}{x^{2} - x - 2}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x + 3}{4 x + 4} \cdot \frac{x^{2} - x - 2}{x^{2} \cdot (2 x) - 3}$

Step 2

Factor $4$ out of $4 x + 4$ .

Tap for more steps...

Step 2.1

Factor $4$ out of $4 x$ .

$\frac{x + 3}{4 (x) + 4} \cdot \frac{x^{2} - x - 2}{x^{2} \cdot (2 x) - 3}$

Step 2.2

Factor $4$ out of $4$ .

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

Step 2.3

Factor $4$ out of $4 (x) + 4 (1)$ .

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

Step 3

Factor $x^{2} - x - 2$ using the AC method.

Tap for more steps...

Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify the denominator.

Tap for more steps...

Step 4.1

Rewrite using the commutative property of multiplication.

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Move $x$ .

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

Step 4.2.2

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

Tap for more steps...

Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.3

Add $1$ and $2$ .

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

Step 5

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 5.1

Factor $x + 1$ out of $4 (x + 1)$ .

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

Step 5.2

Factor $x + 1$ out of $(x - 2) (x + 1)$ .

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

Step 5.3

Cancel the common factor.

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

Step 5.4

Rewrite the expression.

$\frac{x + 3}{4} \cdot \frac{x - 2}{2 x^{3} - 3}$

Step 6

Multiply $\frac{x + 3}{4}$ by $\frac{x - 2}{2 x^{3} - 3}$ .

$\frac{(x + 3) (x - 2)}{4 (2 x^{3} - 3)}$

Step 7

Expand $(x + 3) (x - 2)$ using the FOIL Method.

Tap for more steps...

Step 7.1

Apply the distributive property.

$\frac{x (x - 2) + 3 (x - 2)}{4 (2 x^{3} - 3)}$

Step 7.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 + 3 (x - 2)}{4 (2 x^{3} - 3)}$

Step 7.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 + 3 x + 3 \cdot - 2}{4 (2 x^{3} - 3)}$

Step 8

Simplify and combine like terms.

Tap for more steps...

Step 8.1

Simplify each term.

Tap for more steps...

Step 8.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 2 + 3 x + 3 \cdot - 2}{4 (2 x^{3} - 3)}$

Step 8.1.2

Move $- 2$ to the left of $x$ .

$\frac{x^{2} - 2 \cdot x + 3 x + 3 \cdot - 2}{4 (2 x^{3} - 3)}$

Step 8.1.3

Multiply $3$ by $- 2$ .

$\frac{x^{2} - 2 x + 3 x - 6}{4 (2 x^{3} - 3)}$

Step 8.2

Add $- 2 x$ and $3 x$ .

$\frac{x^{2} + x - 6}{4 (2 x^{3} - 3)}$

Step 9

Split the fraction $\frac{x^{2} + x - 6}{4 (2 x^{3} - 3)}$ into two fractions.

$\frac{x^{2} + x}{4 (2 x^{3} - 3)} + \frac{- 6}{4 (2 x^{3} - 3)}$

Step 10

Split the fraction $\frac{x^{2} + x}{4 (2 x^{3} - 3)}$ into two fractions.

$\frac{x^{2}}{4 (2 x^{3} - 3)} + \frac{x}{4 (2 x^{3} - 3)} + \frac{- 6}{4 (2 x^{3} - 3)}$

Step 11

Cancel the common factor of $- 6$ and $4$ .

Tap for more steps...

Step 11.1

Factor $2$ out of $- 6$ .

$\frac{x^{2}}{4 (2 x^{3} - 3)} + \frac{x}{4 (2 x^{3} - 3)} + \frac{2 (- 3)}{4 (2 x^{3} - 3)}$

Step 11.2

Cancel the common factors.

Tap for more steps...

Step 11.2.1

Factor $2$ out of $4 (2 x^{3} - 3)$ .

$\frac{x^{2}}{4 (2 x^{3} - 3)} + \frac{x}{4 (2 x^{3} - 3)} + \frac{2 (- 3)}{2 (2 (2 x^{3} - 3))}$

Step 11.2.2

Cancel the common factor.

$\frac{x^{2}}{4 (2 x^{3} - 3)} + \frac{x}{4 (2 x^{3} - 3)} + \frac{2 \cdot - 3}{2 (2 (2 x^{3} - 3))}$

Step 11.2.3

Rewrite the expression.

$\frac{x^{2}}{4 (2 x^{3} - 3)} + \frac{x}{4 (2 x^{3} - 3)} + \frac{- 3}{2 (2 x^{3} - 3)}$

Step 12

Move the negative in front of the fraction.

$\frac{x^{2}}{4 (2 x^{3} - 3)} + \frac{x}{4 (2 x^{3} - 3)} - \frac{3}{2 (2 x^{3} - 3)}$