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

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

Step 1

Factor $x^{2} + 4 x - 5$ 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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Write the factored form using these integers.

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

Step 2

To write $\frac{4 x}{(x - 1) (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 3

To write $- \frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{(x - 1) (x + 5)}{(x - 1) (x + 5)}$ .

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

Step 4

Write each expression with a common denominator of $(x - 1) (x + 5) \cdot 4$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(x - 1) (x + 5) \cdot 4$ .

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

Reorder the factors of $4 ((x - 1) (x + 5))$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Multiply $4$ by $4$ .

$\frac{16 x - 5 (x - 1) (x + 5)}{4 (x + 5) (x - 1)}$

Apply the distributive property.

$\frac{16 x + (- 5 x - 5 \cdot - 1) (x + 5)}{4 (x + 5) (x - 1)}$

Multiply $- 5$ by $- 1$ .

$\frac{16 x + (- 5 x + 5) (x + 5)}{4 (x + 5) (x - 1)}$

Expand $(- 5 x + 5) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{16 x - 5 x (x + 5) + 5 (x + 5)}{4 (x + 5) (x - 1)}$

Apply the distributive property.

$\frac{16 x - 5 x \cdot x - 5 x \cdot 5 + 5 (x + 5)}{4 (x + 5) (x - 1)}$

Apply the distributive property.

$\frac{16 x - 5 x \cdot x - 5 x \cdot 5 + 5 x + 5 \cdot 5}{4 (x + 5) (x - 1)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{16 x - 5 (x \cdot x) - 5 x \cdot 5 + 5 x + 5 \cdot 5}{4 (x + 5) (x - 1)}$

Multiply $x$ by $x$ .

$\frac{16 x - 5 x^{2} - 5 x \cdot 5 + 5 x + 5 \cdot 5}{4 (x + 5) (x - 1)}$

Multiply $5$ by $- 5$ .

$\frac{16 x - 5 x^{2} - 25 x + 5 x + 5 \cdot 5}{4 (x + 5) (x - 1)}$

Multiply $5$ by $5$ .

$\frac{16 x - 5 x^{2} - 25 x + 5 x + 25}{4 (x + 5) (x - 1)}$

Add $- 25 x$ and $5 x$ .

$\frac{16 x - 5 x^{2} - 20 x + 25}{4 (x + 5) (x - 1)}$

Subtract $20 x$ from $16 x$ .

$\frac{- 5 x^{2} - 4 x + 25}{4 (x + 5) (x - 1)}$

Step 7

Simplify with factoring out.

Factor $- 1$ out of $- 5 x^{2}$ .

$\frac{- (5 x^{2}) - 4 x + 25}{4 (x + 5) (x - 1)}$

Factor $- 1$ out of $- 4 x$ .

$\frac{- (5 x^{2}) - (4 x) + 25}{4 (x + 5) (x - 1)}$

Factor $- 1$ out of $- (5 x^{2}) - (4 x)$ .

$\frac{- (5 x^{2} + 4 x) + 25}{4 (x + 5) (x - 1)}$

Rewrite $25$ as $- 1 (- 25)$ .

$\frac{- (5 x^{2} + 4 x) - 1 (- 25)}{4 (x + 5) (x - 1)}$

Factor $- 1$ out of $- (5 x^{2} + 4 x) - 1 (- 25)$ .

$\frac{- (5 x^{2} + 4 x - 25)}{4 (x + 5) (x - 1)}$

Simplify the expression.

Rewrite $- (5 x^{2} + 4 x - 25)$ as $- 1 (5 x^{2} + 4 x - 25)$ .

$\frac{- 1 (5 x^{2} + 4 x - 25)}{4 (x + 5) (x - 1)}$

Move the negative in front of the fraction.

$- \frac{5 x^{2} + 4 x - 25}{4 (x + 5) (x - 1)}$

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

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