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

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

Step 1

Factor $x$ out of $x^{2} + 2 x$ .

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

$\frac{x + 5}{x + 2} - \frac{x + 1}{x \cdot x + 2 x}$

Factor $x$ out of $2 x$ .

$\frac{x + 5}{x + 2} - \frac{x + 1}{x \cdot x + x \cdot 2}$

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

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

Step 2

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

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

Step 3

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

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

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

Reorder the factors of $(x + 2) x$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $x$ from $5 x$ .

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