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

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

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

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

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

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

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

Multiply $\frac{x + 3}{2 x^{2} + 1}$ by $\frac{1 - 2 x}{1 - 2 x}$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $1$ .

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

Multiply $- 2$ by $3$ .

$\frac{x - 2 x^{2} + 3 - 6 x - 5 (2 x^{2} + 1)}{(1 - 2 x) (2 x^{2} + 1)}$

Subtract $6 x$ from $x$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 5$ .

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

Multiply $- 5$ by $1$ .

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

Subtract $10 x^{2}$ from $- 2 x^{2}$ .

$\frac{- 5 x - 12 x^{2} + 3 - 5}{(1 - 2 x) (2 x^{2} + 1)}$

Subtract $5$ from $3$ .

$\frac{- 5 x - 12 x^{2} - 2}{(1 - 2 x) (2 x^{2} + 1)}$

Reorder terms.

$\frac{- 12 x^{2} - 5 x - 2}{(1 - 2 x) (2 x^{2} + 1)}$

Simplify with factoring out.

Factor $- 1$ out of $- 12 x^{2}$ .

$\frac{- (12 x^{2}) - 5 x - 2}{(1 - 2 x) (2 x^{2} + 1)}$

Factor $- 1$ out of $- 5 x$ .

$\frac{- (12 x^{2}) - (5 x) - 2}{(1 - 2 x) (2 x^{2} + 1)}$

Factor $- 1$ out of $- (12 x^{2}) - (5 x)$ .

$\frac{- (12 x^{2} + 5 x) - 2}{(1 - 2 x) (2 x^{2} + 1)}$

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- (12 x^{2} + 5 x) - 1 (2)}{(1 - 2 x) (2 x^{2} + 1)}$

Factor $- 1$ out of $- (12 x^{2} + 5 x) - 1 (2)$ .

$\frac{- (12 x^{2} + 5 x + 2)}{(1 - 2 x) (2 x^{2} + 1)}$

Simplify the expression.

Rewrite $- (12 x^{2} + 5 x + 2)$ as $- 1 (12 x^{2} + 5 x + 2)$ .

$\frac{- 1 (12 x^{2} + 5 x + 2)}{(1 - 2 x) (2 x^{2} + 1)}$

Move the negative in front of the fraction.

$- \frac{12 x^{2} + 5 x + 2}{(1 - 2 x) (2 x^{2} + 1)}$

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

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