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

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

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

To write $- \frac{3}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

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

Multiply $\frac{3}{x - 2}$ by $\frac{x + 2}{x + 2}$ .

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 3$ by $2$ .

$\frac{4 x + 1 - 3 x - 6}{(x + 2) (x - 2)}$

Subtract $3 x$ from $4 x$ .

$\frac{x + 1 - 6}{(x + 2) (x - 2)}$

Subtract $6$ from $1$ .

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