Question

select the simplified form of this expression.

\\\frac{15x}{x^2 + 6x + 9} + \frac{5}{x + 3}\\

Explanation:

Factor the denominators

We factor the quadratic expression in the first denominator.

$$ x^2 + 6x + 9 = (x + 3)^2 $$

Find a common denominator

The denominators are \((x + 3)^2\) and \((x + 3)\). The least common denominator is \((x + 3)^2\).
We rewrite the second term with this common denominator:

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

Combine the numerators

We add the numerators over the common denominator.

$$ \frac{15x}{(x + 3)^2} + \frac{5(x + 3)}{(x + 3)^2} = \frac{15x + 5(x + 3)}{(x + 3)^2} $$

Simplify the numerator

We expand and combine like terms in the numerator.

$$ 15x + 5x + 15 = 20x + 15 $$

Thus, the simplified expression is:

$$ \frac{20x + 15}{(x + 3)^2} $$

We can also factor out the greatest common factor in the numerator:

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

Answer:

$$\frac{20x + 15}{(x + 3)^2}$$