Simplify (3x)/(2y+3)+(x^2+3x)/(4xy-3-2y+6x)

$\frac{3 x}{2 y + 3} + \frac{x^{2} + 3 x}{4 x y - 3 - 2 y + 6 x}$

Step 1

Simplify each term.

Factor $x$ out of $x^{2} + 3 x$ .

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

$\frac{3 x}{2 y + 3} + \frac{x \cdot x + 3 x}{4 x y - 3 - 2 y + 6 x}$

Factor $x$ out of $3 x$ .

$\frac{3 x}{2 y + 3} + \frac{x \cdot x + x \cdot 3}{4 x y - 3 - 2 y + 6 x}$

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

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

Simplify the denominator.

Reorder terms.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $2 y + 3$ .

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

Step 2

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

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

Step 3

Simplify terms.

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

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Factor $x$ out of $3 x (2 x - 1) + x (x + 3)$ .

Factor $x$ out of $3 x (2 x - 1)$ .

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

Factor $x$ out of $x (3 (2 x - 1)) + x (x + 3)$ .

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

Apply the distributive property.

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

Multiply $2$ by $3$ .

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

Multiply $3$ by $- 1$ .

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

Add $6 x$ and $x$ .

$\frac{x (7 x - 3 + 3)}{(2 y + 3) (2 x - 1)}$

Add $- 3$ and $3$ .

$\frac{x (7 x + 0)}{(2 y + 3) (2 x - 1)}$

Add $7 x$ and $0$ .

$\frac{x \cdot 7 x}{(2 y + 3) (2 x - 1)}$

Combine exponents.

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

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

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Simplify (3x)/(2y+3)+(x^2+3x)/(4xy-3-2y+6x)

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