To find the product of the expression \((3n + 2)(-4n^2 - 4n + 2)\), we'll use the distributive property (also known as the FOIL method for two binomials), which involves distributing each term in the first factor by each term in the second factor.
Let's perform the multiplication step by step:
-
Distribute \(3n\): \[ 3n \cdot (-4n^2) = -12n^3 \] \[ 3n \cdot (-4n) = -12n^2 \] \[ 3n \cdot (2) = 6n \]
-
Distribute \(2\): \[ 2 \cdot (-4n^2) = -8n^2 \] \[ 2 \cdot (-4n) = -8n \] \[ 2 \cdot (2) = 4 \]
Now, combining all these results together: \[ -12n^3 + (-12n^2) + 6n + (-8n^2) + (-8n) + 4 \]
Next, we can combine like terms:
- For \(n^3\): \(-12n^3\) (only one term)
- For \(n^2\): \(-12n^2 - 8n^2 = -20n^2\)
- For \(n\): \(6n - 8n = -2n\)
- Constant: \(4\)
Putting it all together, we have: \[ -12n^3 - 20n^2 - 2n + 4 \]
Thus, the simplified product of the expression \((3n + 2)(-4n^2 - 4n + 2)\) is: \[ \boxed{-12n^3 - 20n^2 - 2n + 4} \]
1 answer