Simplify 4/(3x^2+15x)+(7x)/(x^2+10x+25)

$\frac{4}{3 x^{2} + 15 x} + \frac{7 x}{x^{2} + 10 x + 25}$

Simplify each term.

Factor $3 x$ out of $3 x^{2} + 15 x$ .

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

$\frac{4}{3 x (x) + 15 x} + \frac{7 x}{x^{2} + 10 x + 25}$

Factor $3 x$ out of $15 x$ .

$\frac{4}{3 x (x) + 3 x (5)} + \frac{7 x}{x^{2} + 10 x + 25}$

Factor $3 x$ out of $3 x (x) + 3 x (5)$ .

$\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 10 x + 25}$

Factor using the perfect square rule.

Rewrite $25$ as $5^{2}$ .

$\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 10 x + 5^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Rewrite the polynomial.

$\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{4}{3 x (x + 5)} + \frac{7 x}{{(x + 5)}^{2}}$

To write $\frac{4}{3 x (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{4}{3 x (x + 5)} \cdot \frac{x + 5}{x + 5} + \frac{7 x}{{(x + 5)}^{2}}$

To write $\frac{7 x}{{(x + 5)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{3 x}{3 x}$ .

$\frac{4}{3 x (x + 5)} \cdot \frac{x + 5}{x + 5} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Write each expression with a common denominator of $3 x {(x + 5)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4}{3 x (x + 5)}$ by $\frac{x + 5}{x + 5}$ .

$\frac{4 (x + 5)}{3 x (x + 5) (x + 5)} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Raise $x + 5$ to the power of $1$ .

$\frac{4 (x + 5)}{3 x ({(x + 5)}^{1} (x + 5))} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Raise $x + 5$ to the power of $1$ .

$\frac{4 (x + 5)}{3 x ({(x + 5)}^{1} {(x + 5)}^{1})} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

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

$\frac{4 (x + 5)}{3 x {(x + 5)}^{1 + 1}} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Add $1$ and $1$ .

$\frac{4 (x + 5)}{3 x {(x + 5)}^{2}} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Multiply $\frac{7 x}{{(x + 5)}^{2}}$ by $\frac{3 x}{3 x}$ .

$\frac{4 (x + 5)}{3 x {(x + 5)}^{2}} + \frac{7 x (3 x)}{{(x + 5)}^{2} (3 x)}$

Reorder the factors of ${(x + 5)}^{2} (3 x)$ .

$\frac{4 (x + 5)}{3 x {(x + 5)}^{2}} + \frac{7 x (3 x)}{3 x {(x + 5)}^{2}}$

Combine the numerators over the common denominator.

$\frac{4 (x + 5) + 7 x (3 x)}{3 x {(x + 5)}^{2}}$

Simplify the numerator.

Apply the distributive property.

$\frac{4 x + 4 \cdot 5 + 7 x (3 x)}{3 x {(x + 5)}^{2}}$

Multiply $4$ by $5$ .

$\frac{4 x + 20 + 7 x (3 x)}{3 x {(x + 5)}^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{4 x + 20 + 7 \cdot 3 x \cdot x}{3 x {(x + 5)}^{2}}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{4 x + 20 + 7 \cdot 3 (x \cdot x)}{3 x {(x + 5)}^{2}}$

Multiply $x$ by $x$ .

$\frac{4 x + 20 + 7 \cdot 3 x^{2}}{3 x {(x + 5)}^{2}}$

Multiply $7$ by $3$ .

$\frac{4 x + 20 + 21 x^{2}}{3 x {(x + 5)}^{2}}$

Reorder terms.

$\frac{21 x^{2} + 4 x + 20}{3 x {(x + 5)}^{2}}$

Simplify 4/(3x^2+15x)+(7x)/(x^2+10x+25)

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