Simplify (x-4)/(x^2+4x+3)-(x-1)/(x^2-9)

$\frac{x - 4}{x^{2} + 4 x + 3} - \frac{x - 1}{x^{2} - 9}$

Step 1

Simplify each term.

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

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 $4$ .

$1, 3$

Write the factored form using these integers.

$\frac{x - 4}{(x + 1) (x + 3)} - \frac{x - 1}{x^{2} - 9}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

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 $b = 3$ .

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

Step 2

To write $\frac{x - 4}{(x + 1) (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(x + 1) (x + 3) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 4}{(x + 1) (x + 3)}$ by $\frac{x - 3}{x - 3}$ .

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

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

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

Reorder the factors of $(x + 3) (x - 3) (x + 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Multiply $- 4$ by $- 3$ .

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

Subtract $4 x$ from $- 3 x$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $- x$ and $x$ .

$\frac{x^{2} - 7 x + 12 - x^{2} + 0 + 1}{(x + 1) (x + 3) (x - 3)}$

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

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

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

$\frac{- 7 x + 12 + 0 + 1}{(x + 1) (x + 3) (x - 3)}$

Add $- 7 x$ and $0$ .

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

Add $12$ and $1$ .

$\frac{- 7 x + 13}{(x + 1) (x + 3) (x - 3)}$

Step 7

Simplify with factoring out.

Factor $- 1$ out of $- 7 x$ .

$\frac{- (7 x) + 13}{(x + 1) (x + 3) (x - 3)}$

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (7 x) - 1 (- 13)}{(x + 1) (x + 3) (x - 3)}$

Factor $- 1$ out of $- (7 x) - 1 (- 13)$ .

$\frac{- (7 x - 13)}{(x + 1) (x + 3) (x - 3)}$

Simplify the expression.

Rewrite $- (7 x - 13)$ as $- 1 (7 x - 13)$ .

$\frac{- 1 (7 x - 13)}{(x + 1) (x + 3) (x - 3)}$

Move the negative in front of the fraction.

$- \frac{7 x - 13}{(x + 1) (x + 3) (x - 3)}$

Simplify (x-4)/(x^2+4x+3)-(x-1)/(x^2-9)

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