To simplify the expression \( \frac{5x}{x^2 + 5x + 6} + \frac{x}{x + 3} \), we first need to factor the denominator of the first term.
The expression \( x^2 + 5x + 6 \) can be factored as:
\[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
Now we can rewrite the expression:
\[ \frac{5x}{(x + 2)(x + 3)} + \frac{x}{x + 3} \]
Next, we need a common denominator to combine the fractions. The common denominator is \( (x + 2)(x + 3) \).
Now we rewrite the second term to have the common denominator:
\[ \frac{x}{x + 3} = \frac{x(x + 2)}{(x + 2)(x + 3)} \]
Now we can combine the two fractions:
\[ \frac{5x}{(x + 2)(x + 3)} + \frac{x(x + 2)}{(x + 2)(x + 3)} = \frac{5x + x(x + 2)}{(x + 2)(x + 3)} \]
Expanding the numerator:
\[ x(x + 2) = x^2 + 2x \]
So now the numerator becomes:
\[ 5x + x^2 + 2x = x^2 + 7x \]
Thus, we have:
\[ \frac{x^2 + 7x}{(x + 2)(x + 3)} \]
Now the expression is fully simplified.
The final result is:
\[ \frac{x^2 + 7x}{x^2 + 5x + 6} \]
To match your original formatting:
\[ \frac{□}{x^2 + 5x + 6} \quad \text{where } □ = x^2 + 7x \]
So the simplified form of the given rational expression is:
\[ \frac{x^2 + 7x}{x^2 + 5x + 6} \]