Question

Simplify the following rational expressions:
x
2
+5x+6
5x
​
+
x+3
x
​
=
x
2
+5x+6

Answer 1

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:

1. 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,
\[
\frac{x^2 + 5x + 6}{5x} = \frac{(x+2)(x+3)}{5x}
\]

2. 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:
1. Start with \((x+2)(x+3) = x^2 + 5x + 6\).
2. 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!