Simplify (2x+3)/(x+4)+(x-1)/(x^2+2x-8)

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

Step 1

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.

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

Step 2

To write $\frac{2 x + 3}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

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

Multiply $\frac{2 x + 3}{x + 4}$ by $\frac{x - 2}{x - 2}$ .

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Expand $(2 x + 3) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $2$ .

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

Multiply $3$ by $- 2$ .

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

Add $- 4 x$ and $3 x$ .

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

Add $- x$ and $x$ .

$\frac{2 x^{2} + 0 - 6 - 1}{(x + 4) (x - 2)}$

Add $2 x^{2}$ and $0$ .

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

Subtract $1$ from $- 6$ .

$\frac{2 x^{2} - 7}{(x + 4) (x - 2)}$

Simplify (2x+3)/(x+4)+(x-1)/(x^2+2x-8)

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