Simplify (2x+4)/(x+1)+5/(x-2)

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

Step 1

Factor $2$ out of $2 x + 4$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $4$ .

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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

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

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

Reorder the factors of $(x - 2) (x + 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $2$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $2$ .

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

Multiply $4$ by $- 2$ .

$\frac{2 x^{2} - 4 x + 4 x - 8 + 5 (x + 1)}{(x + 1) (x - 2)}$

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

$\frac{2 x^{2} + 0 - 8 + 5 (x + 1)}{(x + 1) (x - 2)}$

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

$\frac{2 x^{2} - 8 + 5 (x + 1)}{(x + 1) (x - 2)}$

Apply the distributive property.

$\frac{2 x^{2} - 8 + 5 x + 5 \cdot 1}{(x + 1) (x - 2)}$

Multiply $5$ by $1$ .

$\frac{2 x^{2} - 8 + 5 x + 5}{(x + 1) (x - 2)}$

Add $- 8$ and $5$ .

$\frac{2 x^{2} + 5 x - 3}{(x + 1) (x - 2)}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$\frac{2 x^{2} + 5 (x) - 3}{(x + 1) (x - 2)}$

Rewrite $5$ as $- 1$ plus $6$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

Simplify (2x+4)/(x+1)+5/(x-2)

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