We are asked to simplify the complex fraction $\frac{c^2 + 7c + 10}{\frac{c^2 + 5c}{c^3 + 2c^2}}$.

1. Problem Description

We are asked to simplify the complex fraction $\frac{c^2 + 7c + 10}{\frac{c^2 + 5c}{c^3 + 2c^2}}$.

2. Solution Steps

First, we rewrite the expression as a division problem:

$$$$

\frac{c^2 + 7c + 10}{\frac{c^2 + 5c}{c^3 + 2c^2}} = (c^2 + 7c + 10) \div \frac{c^2 + 5c}{c^3 + 2c^2}

$$$$

Then, we change division to multiplication by the reciprocal:

$$$$

(c^2 + 7c + 10) \cdot \frac{c^3 + 2c^2}{c^2 + 5c}

$$$$

Next, we factor each polynomial:

$$$$

c^2 + 7c + 10 = (c+2)(c+5)

$$$$

$$$$

c^3 + 2c^2 = c^2(c+2)

$$$$

$$$$

c^2 + 5c = c(c+5)

$$$$

Substituting the factored forms, we have:

$$$$

(c+2)(c+5) \cdot \frac{c^2(c+2)}{c(c+5)}

$$$$

Now, we simplify the expression by canceling common factors:

$$$$

\frac{(c+2)(c+5) \cdot c^2(c+2)}{c(c+5)} = \frac{(c+2)(c+5) \cdot c \cdot c(c+2)}{c(c+5)}

$$$$

Cancel out $(c+5)$ and $c$:

$$$$

(c+2) \cdot c(c+2) = c(c+2)^2

$$$$

Expanding the square:

$$$$

c(c+2)^2 = c(c^2 + 4c + 4)

$$$$

Finally, distribute the $c$:

$$$$

c(c^2 + 4c + 4) = c^3 + 4c^2 + 4c

$$$$

3. Final Answer

$c^3 + 4c^2 + 4c$