Simplify (2x+1)/(x-2)+(x-3)/(x^2+2x-8)

$\frac{2 x + 1}{x - 2} + \frac{x - 3}{x^{2} + 2 x - 8}$

Factor $x^{2} + 2 x - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Write the factored form using these integers.

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

To write $\frac{2 x + 1}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

Simplify terms.

Multiply $\frac{2 x + 1}{x - 2}$ by $\frac{x + 4}{x + 4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Expand $(2 x + 1) (x + 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $4$ by $2$ .

$\frac{2 x^{2} + 8 x + 1 x + 1 \cdot 4 + x - 3}{(x - 2) (x + 4)}$

Multiply $x$ by $1$ .

$\frac{2 x^{2} + 8 x + x + 1 \cdot 4 + x - 3}{(x - 2) (x + 4)}$

Multiply $4$ by $1$ .

$\frac{2 x^{2} + 8 x + x + 4 + x - 3}{(x - 2) (x + 4)}$

Add $8 x$ and $x$ .

$\frac{2 x^{2} + 9 x + 4 + x - 3}{(x - 2) (x + 4)}$

Add $9 x$ and $x$ .

$\frac{2 x^{2} + 10 x + 4 - 3}{(x - 2) (x + 4)}$

Subtract $3$ from $4$ .

$\frac{2 x^{2} + 10 x + 1}{(x - 2) (x + 4)}$

Simplify (2x+1)/(x-2)+(x-3)/(x^2+2x-8)

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