To simplify the expression given in your question, it looks like you have a combination of rational expressions. Let's clarify the expression:
You wrote the expression as:
\[ \frac{x^2 + 5x + 6}{5x} + \frac{x + 3}{x} = \frac{x^2 + 5x + 6}{5x} \]
First, let’s focus on simplifying the left-hand side:
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Simplify \(\frac{x^2 + 5x + 6}{5x}\):
- The numerator \(x^2 + 5x + 6\) can be factored: \[ x^2 + 5x + 6 = (x+2)(x+3) \] Thus, \[ \frac{x^2 + 5x + 6}{5x} = \frac{(x+2)(x+3)}{5x} \]
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Now simplify \(\frac{x + 3}{x}\):
- This can be rewritten as: \[ \frac{x + 3}{x} = 1 + \frac{3}{x} \]
So the left-hand side can be rewritten as: \[ \frac{(x+2)(x+3)}{5x} + 1 + \frac{3}{x} \]
To combine these, you need a common denominator. The common denominator would be \(5x\):
- The term \(1\) can be written as \(\frac{5x}{5x}\).
- The term \(\frac{3}{x}\) can be rewritten as \(\frac{15}{5x}\).
Now the expression simplifies to: \[ \frac{(x+2)(x+3) + 5x + 15}{5x} \]
Now we can combine the numerators:
- Start with \((x+2)(x+3) = x^2 + 5x + 6\).
- Add \(5x + 15\): \[ x^2 + 5x + 6 + 5x + 15 = x^2 + 10x + 21 \]
So now the left side is: \[ \frac{x^2 + 10x + 21}{5x} \]
Now let's simplify the left-hand side: \[ x^2 + 10x + 21 = (x + 3)(x + 7) \]
So it becomes: \[ \frac{(x+3)(x+7)}{5x} \]
Now putting it all together, your original expressions both simplify into: \[ \frac{(x + 3)(x + 7)}{5x} \]
Thus, the entire expression is simplified to: \[ \frac{(x+3)(x+7)}{5x} \]
If you have any further questions or need more assistance, feel free to ask!