Simplify (5x)/(x^2+3x-4)-(x+1)/(x-1)

$\frac{5 x}{x^{2} + 3 x - 4} - \frac{x + 1}{x - 1}$

Factor $x^{2} + 3 x - 4$ 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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Write the factored form using these integers.

$\frac{5 x}{(x - 1) (x + 4)} - \frac{x + 1}{x - 1}$

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

$\frac{5 x}{(x - 1) (x + 4)} - \frac{x + 1}{x - 1} \cdot \frac{x + 4}{x + 4}$

Simplify terms.

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

$\frac{5 x}{(x - 1) (x + 4)} - \frac{(x + 1) (x + 4)}{(x - 1) (x + 4)}$

Combine the numerators over the common denominator.

$\frac{5 x - (x + 1) (x + 4)}{(x - 1) (x + 4)}$

Simplify the numerator.

Apply the distributive property.

$\frac{5 x + (- x - 1 \cdot 1) (x + 4)}{(x - 1) (x + 4)}$

Multiply $- 1$ by $1$ .

$\frac{5 x + (- x - 1) (x + 4)}{(x - 1) (x + 4)}$

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

Apply the distributive property.

$\frac{5 x - x (x + 4) - 1 (x + 4)}{(x - 1) (x + 4)}$

Apply the distributive property.

$\frac{5 x - x \cdot x - x \cdot 4 - 1 (x + 4)}{(x - 1) (x + 4)}$

Apply the distributive property.

$\frac{5 x - x \cdot x - x \cdot 4 - 1 x - 1 \cdot 4}{(x - 1) (x + 4)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{5 x - (x \cdot x) - x \cdot 4 - 1 x - 1 \cdot 4}{(x - 1) (x + 4)}$

Multiply $x$ by $x$ .

$\frac{5 x - x^{2} - x \cdot 4 - 1 x - 1 \cdot 4}{(x - 1) (x + 4)}$

Multiply $4$ by $- 1$ .

$\frac{5 x - x^{2} - 4 x - 1 x - 1 \cdot 4}{(x - 1) (x + 4)}$

Rewrite $- 1 x$ as $- x$ .

$\frac{5 x - x^{2} - 4 x - x - 1 \cdot 4}{(x - 1) (x + 4)}$

Multiply $- 1$ by $4$ .

$\frac{5 x - x^{2} - 4 x - x - 4}{(x - 1) (x + 4)}$

Subtract $x$ from $- 4 x$ .

$\frac{5 x - x^{2} - 5 x - 4}{(x - 1) (x + 4)}$

Subtract $5 x$ from $5 x$ .

$\frac{- x^{2} + 0 - 4}{(x - 1) (x + 4)}$

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

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

Simplify with factoring out.

Factor $- 1$ out of $- x^{2}$ .

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

Rewrite $- 4$ as $- 1 (4)$ .

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

Factor $- 1$ out of $- (x^{2}) - 1 (4)$ .

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

Simplify the expression.

Rewrite $- (x^{2} + 4)$ as $- 1 (x^{2} + 4)$ .

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

Move the negative in front of the fraction.

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

Simplify (5x)/(x^2+3x-4)-(x+1)/(x-1)

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