The expression provided in the image is:

$\frac{10p + 20}{3p} + \frac{5p - 10}{p^3}$

Let's solve this step by step.

Step 1: Simplify each fraction

Simplifying the first fraction:

$\frac{10p + 20}{3p}$ Factor out the common factor from the numerator: $\frac{10(p + 2)}{3p}$ This expression cannot be simplified further as there are no common factors between the numerator and the denominator.

Simplifying the second fraction:

$\frac{5p - 10}{p^3}$ Factor out the common factor from the numerator: $\frac{5(p - 2)}{p^3}$ This expression also cannot be simplified further.

Step 2: Find a common denominator

To add these two fractions, we need a common denominator. The denominators are $3p$ and $p^3$. The least common denominator (LCD) is $3p^3$.

Convert each fraction to have the LCD:

1. For $\frac{10(p + 2)}{3p}$: $\frac{10(p + 2)}{3p} = \frac{10(p + 2) \cdot p^2}{3p \cdot p^2} = \frac{10p^2(p + 2)}{3p^3}$

2. For $\frac{5(p - 2)}{p^3}$: $\frac{5(p - 2)}{p^3} = \frac{5(p - 2) \cdot 3}{p^3 \cdot 3} = \frac{15(p - 2)}{3p^3}$

Step 3: Add the fractions

Now that both fractions have the same denominator, we can add them:

$\frac{10p^2(p + 2)}{3p^3} + \frac{15(p - 2)}{3p^3} = \frac{10p^2(p + 2) + 15(p - 2)}{3p^3}$

Step 4: Expand and combine like terms

Expand the numerators: $10p^2(p + 2) = 10p^3 + 20p^2$ $15(p - 2) = 15p - 30$

Combine them: $10p^3 + 20p^2 + 15p - 30$

So, the expression becomes: $\frac{10p^3 + 20p^2 + 15p - 30}{3p^3}$

Step 5: Simplify if possible

Now, let's see if the numerator can be factored: $10p^3 + 20p^2 + 15p - 30$ We can factor the numerator: $5(2p^3 + 4p^2 + 3p - 6)$

However, it looks like no further simplification is possible. Therefore, the final expression is: $\frac{5(2p^3 + 4p^2 + 3p - 6)}{3p^3}$

This is the simplified form of the given expression.

Would you like further details or have any questions?

Here are 5 related questions you might explore next:

1. How can this method be applied to add fractions with different denominators in general?
2. What happens if the common denominator isn't the least common denominator?
3. How would the process change if the expression involved subtraction instead of addition?
4. Can you find a situation where the expression simplifies completely (i.e., no p's left in the denominator)?
5. How can this expression be used to solve for specific values of $p$?

Tip: Always factor the numerator and denominator completely before trying to simplify a rational expression. This can often reveal further simplifications.