Trigonometry Examples

$\frac{2 x + 4}{x + 1} - \frac{7 x - 1}{x^{2} + 4 x + 3}$

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $2 x + 4$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 4}{x + 1} - \frac{7 x - 1}{x^{2} + 4 x + 3}$

Step 1.1.2

Factor $2$ out of $4$ .

$\frac{2 x + 2 \cdot 2}{x + 1} - \frac{7 x - 1}{x^{2} + 4 x + 3}$

Step 1.1.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

$\frac{2 (x + 2)}{x + 1} - \frac{7 x - 1}{x^{2} + 4 x + 3}$

Step 1.2

Factor $x^{2} + 4 x + 3$ using the AC method.

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Step 1.2.1

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 $3$ and whose sum is $4$ .

$1, 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{2 (x + 2)}{x + 1} - \frac{7 x - 1}{(x + 1) (x + 3)}$

Step 2

To write $\frac{2 (x + 2)}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{2 (x + 2)}{x + 1} \cdot \frac{x + 3}{x + 3} - \frac{7 x - 1}{(x + 1) (x + 3)}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{2 (x + 2)}{x + 1}$ by $\frac{x + 3}{x + 3}$ .

$\frac{2 (x + 2) (x + 3)}{(x + 1) (x + 3)} - \frac{7 x - 1}{(x + 1) (x + 3)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{2 (x + 2) (x + 3) - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{(2 x + 2 \cdot 2) (x + 3) - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4.2

Multiply $2$ by $2$ .

$\frac{(2 x + 4) (x + 3) - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4.3

Expand $(2 x + 4) (x + 3)$ using the FOIL Method.

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Step 4.3.1

Apply the distributive property.

$\frac{2 x (x + 3) + 4 (x + 3) - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4.3.2

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot 3 + 4 (x + 3) - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4.3.3

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot 3 + 4 x + 4 \cdot 3 - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4.4

Simplify and combine like terms.

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

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

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Step 4.4.1.1.1

Move $x$ .

$\frac{2 (x \cdot x) + 2 x \cdot 3 + 4 x + 4 \cdot 3 - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4.4.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot 3 + 4 x + 4 \cdot 3 - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4.4.1.2

Multiply $3$ by $2$ .

$\frac{2 x^{2} + 6 x + 4 x + 4 \cdot 3 - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4.4.1.3

Multiply $4$ by $3$ .

$\frac{2 x^{2} + 6 x + 4 x + 12 - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4.4.2

Add $6 x$ and $4 x$ .

$\frac{2 x^{2} + 10 x + 12 - (7 x - 1)}{(x + 1) (x + 3)}$

Step 4.5

Apply the distributive property.

$\frac{2 x^{2} + 10 x + 12 - (7 x) - - 1}{(x + 1) (x + 3)}$

Step 4.6

Multiply $7$ by $- 1$ .

$\frac{2 x^{2} + 10 x + 12 - 7 x - - 1}{(x + 1) (x + 3)}$

Step 4.7

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{2} + 10 x + 12 - 7 x + 1}{(x + 1) (x + 3)}$

Step 4.8

Subtract $7 x$ from $10 x$ .

$\frac{2 x^{2} + 3 x + 12 + 1}{(x + 1) (x + 3)}$

Step 4.9

Add $12$ and $1$ .

$\frac{2 x^{2} + 3 x + 13}{(x + 1) (x + 3)}$