Simplify (2x+1)/(x-5)-4/(x^2-3x-10)

$\frac{2 x + 1}{x - 5} - \frac{4}{x^{2} - 3 x - 10}$

Step 1

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

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

Step 2

To write $\frac{2 x + 1}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 3

Simplify terms.

Multiply $\frac{2 x + 1}{x - 5}$ by $\frac{x + 2}{x + 2}$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $2$ .

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

Multiply $x$ by $1$ .

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

Multiply $2$ by $1$ .

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

Add $4 x$ and $x$ .

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

Subtract $4$ from $2$ .

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

Simplify (2x+1)/(x-5)-4/(x^2-3x-10)

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