Pre-Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor by grouping.

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Step 1.1.1

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 $b = 5$ .

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Step 1.1.1.1

Factor $5$ out of $5 x$ .

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

Step 1.1.1.2

Rewrite $5$ as $2$ plus $3$

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

Step 1.1.1.3

Apply the distributive property.

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

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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

Step 1.2

Factor $3$ out of $3 x + 3$ .

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Step 1.2.1

Factor $3$ out of $3 x$ .

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

Step 1.2.2

Factor $3$ out of $3$ .

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

Step 1.2.3

Factor $3$ out of $3 (x) + 3 (1)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(x + 1) (2 x + 3) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x + 1) (2 x + 3) \cdot 3$ .

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

Step 4.4

Reorder the factors of $3 (x + 1) (2 x + 3)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $3$ by $2$ .

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

Step 6.3

Multiply $- 1$ by $3$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $2$ by $2$ .

$\frac{6 x - 3 + 4 x + 2 \cdot 3}{3 (2 x + 3) (x + 1)}$

Step 6.6

Multiply $2$ by $3$ .

$\frac{6 x - 3 + 4 x + 6}{3 (2 x + 3) (x + 1)}$

Step 6.7

Add $6 x$ and $4 x$ .

$\frac{10 x - 3 + 6}{3 (2 x + 3) (x + 1)}$

Step 6.8

Add $- 3$ and $6$ .

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