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

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

Multiply the numerator and denominator of the complex fraction by $3 x^{2}$ .

Multiply $\frac{\frac{x + 4}{3} + \frac{1}{x}}{\frac{3}{x} + \frac{3}{x^{2}}}$ by $\frac{3 x^{2}}{3 x^{2}}$ .

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x^{2} (x + 4) + 3 x}{3 x^{2} \frac{3}{x} + 3 x^{2} \frac{3}{x^{2}}}$

Cancel the common factor of $x$ .

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

$\frac{x^{2} (x + 4) + 3 x}{x (3 x) \frac{3}{x} + 3 x^{2} \frac{3}{x^{2}}}$

Cancel the common factor.

$\frac{x^{2} (x + 4) + 3 x}{x (3 x) \frac{3}{x} + 3 x^{2} \frac{3}{x^{2}}}$

Rewrite the expression.

$\frac{x^{2} (x + 4) + 3 x}{3 x \cdot 3 + 3 x^{2} \frac{3}{x^{2}}}$

Multiply $3$ by $3$ .

$\frac{x^{2} (x + 4) + 3 x}{9 x + 3 x^{2} \frac{3}{x^{2}}}$

Cancel the common factor of $x^{2}$ .

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

$\frac{x^{2} (x + 4) + 3 x}{9 x + x^{2} \cdot 3 \frac{3}{x^{2}}}$

Cancel the common factor.

$\frac{x^{2} (x + 4) + 3 x}{9 x + x^{2} \cdot 3 \frac{3}{x^{2}}}$

Rewrite the expression.

$\frac{x^{2} (x + 4) + 3 x}{9 x + 3 \cdot 3}$

Multiply $3$ by $3$ .

$\frac{x^{2} (x + 4) + 3 x}{9 x + 9}$

Simplify the numerator.

Factor $x$ out of $x^{2} (x + 4) + 3 x$ .

Factor $x$ out of $x^{2} (x + 4)$ .

$\frac{x (x (x + 4)) + 3 x}{9 x + 9}$

Factor $x$ out of $3 x$ .

$\frac{x (x (x + 4)) + x \cdot 3}{9 x + 9}$

Factor $x$ out of $x (x (x + 4)) + x \cdot 3$ .

$\frac{x (x (x + 4) + 3)}{9 x + 9}$

Apply the distributive property.

$\frac{x (x \cdot x + x \cdot 4 + 3)}{9 x + 9}$

Multiply $x$ by $x$ .

$\frac{x (x^{2} + x \cdot 4 + 3)}{9 x + 9}$

Move $4$ to the left of $x$ .

$\frac{x (x^{2} + 4 \cdot x + 3)}{9 x + 9}$

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.

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

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

Simplify terms.

Factor $9$ out of $9 x + 9$ .

Factor $9$ out of $9 x$ .

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

Factor $9$ out of $9$ .

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

Factor $9$ out of $9 (x) + 9 (1)$ .

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

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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