Let's break down and simplify the given expression:

$B = \frac{(x+2)^2}{x} \cdot \left( 1 - \frac{x^2}{x+2} \right) - \frac{x^2 + 6x + 4}{x}$

Step 1: Simplify each part of the expression

1. Simplifying the first part:
   $\frac{(x+2)^2}{x} \cdot \left( 1 - \frac{x^2}{x+2} \right)$

   • First, simplify the term inside the parentheses: $1 - \frac{x^2}{x+2} = \frac{(x+2) - x^2}{x+2} = \frac{x + 2 - x^2}{x+2}$ which simplifies to: $\frac{2 - x^2 + x}{x+2}$
   • Now multiply it by $\frac{(x+2)^2}{x}$: $\frac{(x+2)^2}{x} \cdot \frac{2 - x^2 + x}{x+2}$ Notice that $(x+2)$ cancels out, so the expression simplifies to: $\frac{(x+2)(2 - x^2 + x)}{x}$

2. Simplifying the second part:
   $- \frac{x^2 + 6x + 4}{x}$ This term is already in a simplified form.

Step 2: Combine both parts

Now combine the two parts:

$B = \frac{(x+2)(2 - x^2 + x)}{x} - \frac{x^2 + 6x + 4}{x}$

Both terms have a denominator of $x$, so we can combine them:

$B = \frac{(x+2)(2 - x^2 + x) - (x^2 + 6x + 4)}{x}$

Step 3: Expand the numerator

Expand the first term in the numerator:

$(x+2)(2 - x^2 + x) = x(2 - x^2 + x) + 2(2 - x^2 + x)$ $= x(2) - x(x^2) + x(x) + 2(2) - 2(x^2) + 2(x)$ $= 2x - x^3 + x^2 + 4 - 2x^2 + 2x$ Simplify the result: $= -x^3 + x^2 - 2x^2 + 2x + 2x + 4$ $= -x^3 - x^2 + 4x + 4$

Now subtract $(x^2 + 6x + 4)$ from this:

$(-x^3 - x^2 + 4x + 4) - (x^2 + 6x + 4)$ $= -x^3 - x^2 + 4x + 4 - x^2 - 6x - 4$ $= -x^3 - 2x^2 - 2x$

Step 4: Final expression

Now, place this simplified numerator over $x$:

$B = \frac{-x^3 - 2x^2 - 2x}{x}$

Finally, simplify each term:

$B = -x^2 - 2x - 2$

So, the simplified expression for $B$ is:

$B = -x^2 - 2x - 2$

Would you like more details or further steps? Here are some related questions to explore:

1. How do you factor the expression $-x^2 - 2x - 2$?
2. What happens to $B$ when $x = 0$?
3. How can you find the roots of the quadratic expression $-x^2 - 2x - 2$?
4. How can you graph the function $B = -x^2 - 2x - 2$?
5. What would the expression $B$ look like if the term $(x+2)$ was replaced with $(x-2)$?

Tip: Always check for possible factorizations when you reach a simplified polynomial—sometimes it reveals additional insights!