Simplify 8/(2x^2-7x+5)+(3x+2)/(2x^2-x-10)

$\frac{8}{2 x^{2} - 7 x + 5} + \frac{3 x + 2}{2 x^{2} - x - 10}$

Simplify each term.

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 $a \cdot c = 2 \cdot 5 = 10$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 x$ .

$\frac{8}{2 x^{2} - 7 (x) + 5} + \frac{3 x + 2}{2 x^{2} - x - 10}$

Rewrite $- 7$ as $- 2$ plus $- 5$

$\frac{8}{2 x^{2} + (- 2 - 5) x + 5} + \frac{3 x + 2}{2 x^{2} - x - 10}$

Apply the distributive property.

$\frac{8}{2 x^{2} - 2 x - 5 x + 5} + \frac{3 x + 2}{2 x^{2} - x - 10}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{8}{(2 x^{2} - 2 x) - 5 x + 5} + \frac{3 x + 2}{2 x^{2} - x - 10}$

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

$\frac{8}{2 x (x - 1) - 5 (x - 1)} + \frac{3 x + 2}{2 x^{2} - x - 10}$

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

$\frac{8}{(x - 1) (2 x - 5)} + \frac{3 x + 2}{2 x^{2} - x - 10}$

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 $a \cdot c = 2 \cdot - 10 = - 20$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$\frac{8}{(x - 1) (2 x - 5)} + \frac{3 x + 2}{2 x^{2} - (x) - 10}$

Rewrite $- 1$ as $4$ plus $- 5$

$\frac{8}{(x - 1) (2 x - 5)} + \frac{3 x + 2}{2 x^{2} + (4 - 5) x - 10}$

Apply the distributive property.

$\frac{8}{(x - 1) (2 x - 5)} + \frac{3 x + 2}{2 x^{2} + 4 x - 5 x - 10}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{8}{(x - 1) (2 x - 5)} + \frac{3 x + 2}{(2 x^{2} + 4 x) - 5 x - 10}$

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

$\frac{8}{(x - 1) (2 x - 5)} + \frac{3 x + 2}{2 x (x + 2) - 5 (x + 2)}$

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

$\frac{8}{(x - 1) (2 x - 5)} + \frac{3 x + 2}{(x + 2) (2 x - 5)}$

To write $\frac{8}{(x - 1) (2 x - 5)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{8}{(x - 1) (2 x - 5)} \cdot \frac{x + 2}{x + 2} + \frac{3 x + 2}{(x + 2) (2 x - 5)}$

To write $\frac{3 x + 2}{(x + 2) (2 x - 5)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{8}{(x - 1) (2 x - 5)} \cdot \frac{x + 2}{x + 2} + \frac{3 x + 2}{(x + 2) (2 x - 5)} \cdot \frac{x - 1}{x - 1}$

Write each expression with a common denominator of $(x - 1) (2 x - 5) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{8}{(x - 1) (2 x - 5)}$ by $\frac{x + 2}{x + 2}$ .

$\frac{8 (x + 2)}{(x - 1) (2 x - 5) (x + 2)} + \frac{3 x + 2}{(x + 2) (2 x - 5)} \cdot \frac{x - 1}{x - 1}$

Multiply $\frac{3 x + 2}{(x + 2) (2 x - 5)}$ by $\frac{x - 1}{x - 1}$ .

$\frac{8 (x + 2)}{(x - 1) (2 x - 5) (x + 2)} + \frac{(3 x + 2) (x - 1)}{(x + 2) (2 x - 5) (x - 1)}$

Reorder the factors of $(x - 1) (2 x - 5) (x + 2)$ .

$\frac{8 (x + 2)}{(2 x - 5) (x + 2) (x - 1)} + \frac{(3 x + 2) (x - 1)}{(x + 2) (2 x - 5) (x - 1)}$

Reorder the factors of $(x + 2) (2 x - 5) (x - 1)$ .

$\frac{8 (x + 2)}{(2 x - 5) (x + 2) (x - 1)} + \frac{(3 x + 2) (x - 1)}{(2 x - 5) (x + 2) (x - 1)}$

Combine the numerators over the common denominator.

$\frac{8 (x + 2) + (3 x + 2) (x - 1)}{(2 x - 5) (x + 2) (x - 1)}$

Simplify the numerator.

Apply the distributive property.

$\frac{8 x + 8 \cdot 2 + (3 x + 2) (x - 1)}{(2 x - 5) (x + 2) (x - 1)}$

Multiply $8$ by $2$ .

$\frac{8 x + 16 + (3 x + 2) (x - 1)}{(2 x - 5) (x + 2) (x - 1)}$

Expand $(3 x + 2) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{8 x + 16 + 3 x (x - 1) + 2 (x - 1)}{(2 x - 5) (x + 2) (x - 1)}$

Apply the distributive property.

$\frac{8 x + 16 + 3 x \cdot x + 3 x \cdot - 1 + 2 (x - 1)}{(2 x - 5) (x + 2) (x - 1)}$

Apply the distributive property.

$\frac{8 x + 16 + 3 x \cdot x + 3 x \cdot - 1 + 2 x + 2 \cdot - 1}{(2 x - 5) (x + 2) (x - 1)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{8 x + 16 + 3 (x \cdot x) + 3 x \cdot - 1 + 2 x + 2 \cdot - 1}{(2 x - 5) (x + 2) (x - 1)}$

Multiply $x$ by $x$ .

$\frac{8 x + 16 + 3 x^{2} + 3 x \cdot - 1 + 2 x + 2 \cdot - 1}{(2 x - 5) (x + 2) (x - 1)}$

Multiply $- 1$ by $3$ .

$\frac{8 x + 16 + 3 x^{2} - 3 x + 2 x + 2 \cdot - 1}{(2 x - 5) (x + 2) (x - 1)}$

Multiply $2$ by $- 1$ .

$\frac{8 x + 16 + 3 x^{2} - 3 x + 2 x - 2}{(2 x - 5) (x + 2) (x - 1)}$

Add $- 3 x$ and $2 x$ .

$\frac{8 x + 16 + 3 x^{2} - x - 2}{(2 x - 5) (x + 2) (x - 1)}$

Subtract $x$ from $8 x$ .

$\frac{7 x + 16 + 3 x^{2} - 2}{(2 x - 5) (x + 2) (x - 1)}$

Subtract $2$ from $16$ .

$\frac{7 x + 3 x^{2} + 14}{(2 x - 5) (x + 2) (x - 1)}$

Reorder terms.

$\frac{3 x^{2} + 7 x + 14}{(2 x - 5) (x + 2) (x - 1)}$

Simplify 8/(2x^2-7x+5)+(3x+2)/(2x^2-x-10)

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