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

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Factor $x^{2} + 4 x + 3$ 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 $3$ and whose sum is $4$ .

$1, 3$

Write the factored form using these integers.

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

Step 3

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 $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 x$ .

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

Rewrite $- 3$ as $- 1$ plus $- 2$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 4

Factor $x$ out of $x^{2} + x$ .

Factor $x$ out of $x^{2}$ .

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

Raise $x$ to the power of $1$ .

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

Factor $x$ out of $x^{1}$ .

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

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 5

Cancel the common factor of $x + 1$ .

Factor $x + 1$ out of $x (x + 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Cancel the common factor of $2 x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

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

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

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

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