Simplify 5/(x-4)-(x-6)/(2x+5)

$\frac{5}{x - 4} - \frac{x - 6}{2 x + 5}$

To write $\frac{5}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{2 x + 5}{2 x + 5}$ .

$\frac{5}{x - 4} \cdot \frac{2 x + 5}{2 x + 5} - \frac{x - 6}{2 x + 5}$

To write $- \frac{x - 6}{2 x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{5}{x - 4} \cdot \frac{2 x + 5}{2 x + 5} - \frac{x - 6}{2 x + 5} \cdot \frac{x - 4}{x - 4}$

Write each expression with a common denominator of $(x - 4) (2 x + 5)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{x - 4}$ by $\frac{2 x + 5}{2 x + 5}$ .

$\frac{5 (2 x + 5)}{(x - 4) (2 x + 5)} - \frac{x - 6}{2 x + 5} \cdot \frac{x - 4}{x - 4}$

Multiply $\frac{x - 6}{2 x + 5}$ by $\frac{x - 4}{x - 4}$ .

$\frac{5 (2 x + 5)}{(x - 4) (2 x + 5)} - \frac{(x - 6) (x - 4)}{(2 x + 5) (x - 4)}$

Reorder the factors of $(x - 4) (2 x + 5)$ .

$\frac{5 (2 x + 5)}{(2 x + 5) (x - 4)} - \frac{(x - 6) (x - 4)}{(2 x + 5) (x - 4)}$

$\frac{5 (2 x + 5)}{(2 x + 5) (x - 4)} - \frac{(x - 6) (x - 4)}{(2 x + 5) (x - 4)}$

Combine the numerators over the common denominator.

$\frac{5 (2 x + 5) - (x - 6) (x - 4)}{(2 x + 5) (x - 4)}$

Simplify the numerator.

Apply the distributive property.

$\frac{5 (2 x) + 5 \cdot 5 - (x - 6) (x - 4)}{(2 x + 5) (x - 4)}$

Multiply $2$ by $5$ .

$\frac{10 x + 5 \cdot 5 - (x - 6) (x - 4)}{(2 x + 5) (x - 4)}$

Multiply $5$ by $5$ .

$\frac{10 x + 25 - (x - 6) (x - 4)}{(2 x + 5) (x - 4)}$

Apply the distributive property.

$\frac{10 x + 25 + (- x - - 6) (x - 4)}{(2 x + 5) (x - 4)}$

Multiply $- 1$ by $- 6$ .

$\frac{10 x + 25 + (- x + 6) (x - 4)}{(2 x + 5) (x - 4)}$

Expand $(- x + 6) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{10 x + 25 - x (x - 4) + 6 (x - 4)}{(2 x + 5) (x - 4)}$

Apply the distributive property.

$\frac{10 x + 25 - x \cdot x - x \cdot - 4 + 6 (x - 4)}{(2 x + 5) (x - 4)}$

Apply the distributive property.

$\frac{10 x + 25 - x \cdot x - x \cdot - 4 + 6 x + 6 \cdot - 4}{(2 x + 5) (x - 4)}$

$\frac{10 x + 25 - x \cdot x - x \cdot - 4 + 6 x + 6 \cdot - 4}{(2 x + 5) (x - 4)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{10 x + 25 - (x \cdot x) - x \cdot - 4 + 6 x + 6 \cdot - 4}{(2 x + 5) (x - 4)}$

Multiply $x$ by $x$ .

$\frac{10 x + 25 - x^{2} - x \cdot - 4 + 6 x + 6 \cdot - 4}{(2 x + 5) (x - 4)}$

$\frac{10 x + 25 - x^{2} - x \cdot - 4 + 6 x + 6 \cdot - 4}{(2 x + 5) (x - 4)}$

Multiply $- 4$ by $- 1$ .

$\frac{10 x + 25 - x^{2} + 4 x + 6 x + 6 \cdot - 4}{(2 x + 5) (x - 4)}$

Multiply $6$ by $- 4$ .

$\frac{10 x + 25 - x^{2} + 4 x + 6 x - 24}{(2 x + 5) (x - 4)}$

$\frac{10 x + 25 - x^{2} + 4 x + 6 x - 24}{(2 x + 5) (x - 4)}$

Add $4 x$ and $6 x$ .

$\frac{10 x + 25 - x^{2} + 10 x - 24}{(2 x + 5) (x - 4)}$

$\frac{10 x + 25 - x^{2} + 10 x - 24}{(2 x + 5) (x - 4)}$

Add $10 x$ and $10 x$ .

$\frac{20 x + 25 - x^{2} - 24}{(2 x + 5) (x - 4)}$

Subtract $24$ from $25$ .

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

Reorder terms.

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

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

Simplify with factoring out.

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

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

Factor $- 1$ out of $20 x$ .

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

Factor $- 1$ out of $- (x^{2}) - (- 20 x)$ .

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

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

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

Factor $- 1$ out of $- (x^{2} - 20 x) - 1 (- 1)$ .

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

Simplify the expression.

Rewrite $- (x^{2} - 20 x - 1)$ as $- 1 (x^{2} - 20 x - 1)$ .

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

Move the negative in front of the fraction.

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

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

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

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