Simplify 6/(x^2)+(2x)/(x^2+3)

$\frac{6}{x^{2}} + \frac{2 x}{x^{2} + 3}$

Step 1

To write $\frac{6}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 3}{x^{2} + 3}$ .

$\frac{6}{x^{2}} \cdot \frac{x^{2} + 3}{x^{2} + 3} + \frac{2 x}{x^{2} + 3}$

Step 2

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

$\frac{6}{x^{2}} \cdot \frac{x^{2} + 3}{x^{2} + 3} + \frac{2 x}{x^{2} + 3} \cdot \frac{x^{2}}{x^{2}}$

Step 3

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

Multiply $\frac{6}{x^{2}}$ by $\frac{x^{2} + 3}{x^{2} + 3}$ .

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

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

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

Reorder the factors of $(x^{2} + 3) x^{2}$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Factor $2$ out of $6 (x^{2} + 3) + 2 x \cdot x^{2}$ .

Factor $2$ out of $6 (x^{2} + 3)$ .

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

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

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

Factor $2$ out of $2 (3 (x^{2} + 3)) + 2 (x \cdot x^{2})$ .

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

Apply the distributive property.

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

Multiply $3$ by $3$ .

$\frac{2 (3 x^{2} + 9 + x \cdot x^{2})}{x^{2} (x^{2} + 3)}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

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

$\frac{2 (3 x^{2} + 9 + x^{1} x^{2})}{x^{2} (x^{2} + 3)}$

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

$\frac{2 (3 x^{2} + 9 + x^{1 + 2})}{x^{2} (x^{2} + 3)}$

Add $1$ and $2$ .

$\frac{2 (3 x^{2} + 9 + x^{3})}{x^{2} (x^{2} + 3)}$

Reorder terms.

$\frac{2 (x^{3} + 3 x^{2} + 9)}{x^{2} (x^{2} + 3)}$