Simplify a(b+c)(-b-c) Using the Distributive Property

To simplify the expression $a(b+c)(-b-c)$ using the distributive property, follow these steps:

• Step 1: Apply Distributive Property to $(b+c)(-b-c)$:
  The expression $(b+c)(-b-c)$ can be expanded using the distributive property:
  $(b+c)(-b-c) = b(-b-c) + c(-b-c)$.
• Step 2: Simplify Each Part:
  Let's simplify each term individually:
  - $b(-b) = -b^2$
  - $b(-c) = -bc$
  - $c(-b) = -bc$
  - $c(-c) = -c^2$
  So, combining these results:
  $(b+c)(-b-c) = -b^2 - bc - bc - c^2 = -b^2 - 2bc - c^2$.
• Step 3: Distribute $a$ Over the Result:
  Now, apply a further distribution of $a$ to get:
  $a(-b^2 - 2bc - c^2) = a(-b^2) + a(-2bc) + a(-c^2)$.
• Step 4: Simplify:
  Perform the distribution:
  - $a(-b^2) = -ab^2$
  - $a(-2bc) = -2abc$
  - $a(-c^2) = -ac^2$
  Thus, the expression simplifies to:
  $-ab^2 - 2abc - ac^2$.

Therefore, the simplified expression using the distributive property is $-ab^2 - 2abc - ac^2$.

Given the multiple-choice options, the correct choice that corresponds to our derived expression is:

Choice 4: Yes, $-ab^2 - 2abc - ac^2$