Let's simplify the expression step by step.

Given expression:

$9(5x + 1)\frac{3\sqrt{x^2 + 1}}{1} + \frac{(5x + 1)^5}{2\sqrt{x^2 + 1}}$

Step 1: Get a common denominator

We observe that the terms involve different denominators: $1$ and $2\sqrt{x^2 + 1}$. The least common denominator (LCD) between them is $2\sqrt{x^2 + 1}$.

1. For the first term, $9(5x + 1)\frac{3\sqrt{x^2 + 1}}{1}$, we multiply both the numerator and the denominator by $2\sqrt{x^2 + 1}$ to match the LCD: $9(5x + 1)\frac{3\sqrt{x^2 + 1}}{1} = \frac{9(5x + 1)3\sqrt{x^2 + 1} \cdot 2\sqrt{x^2 + 1}}{2\sqrt{x^2 + 1}} = \frac{18(5x + 1)(x^2 + 1)}{2\sqrt{x^2 + 1}}$

2. The second term, $\frac{(5x + 1)^5}{2\sqrt{x^2 + 1}}$, already has the same denominator, so we leave it as it is.

Step 2: Combine the two terms over the common denominator

Now that both terms have the same denominator, we can combine them: $\frac{18(5x + 1)(x^2 + 1) + (5x + 1)^5}{2\sqrt{x^2 + 1}}$

Step 3: Pull out any common factors in the numerator

We can factor out $(5x + 1)$ from both terms in the numerator: $\frac{(5x + 1)\left[18(x^2 + 1) + (5x + 1)^4\right]}{2\sqrt{x^2 + 1}}$

Step 4: Expand and combine like terms

1. Expand $18(x^2 + 1)$: $18(x^2 + 1) = 18x^2 + 18$

2. Keep $(5x + 1)^4$ as it is, as it's a higher power term.

So, the expression becomes: $\frac{(5x + 1)\left[18x^2 + 18 + (5x + 1)^4\right]}{2\sqrt{x^2 + 1}}$

This is the simplified form. It involves pulling out the common factor and combining the two terms over a common denominator.

Would you like further details on any of the steps?

Here are some questions to explore further:

1. How do you expand the binomial $(5x + 1)^4$?
2. What is the significance of factoring out common terms?
3. Can this expression be simplified further by substituting a specific value for $x$?
4. How do you deal with square roots in the denominator in general?
5. Could you graph the resulting function to understand its behavior?

Tip: When simplifying expressions, always look for opportunities to factor common terms before expanding—it can save time and avoid mistakes.