Expanding (x + 2y + 4z)²
The expression (x + 2y + 4z)² represents the square of a trinomial. To solve it, we can apply the distributive property, also known as FOIL (First, Outer, Inner, Last) method, twice.
Here's the breakdown:
1. Expanding the First Pair:

We begin by expanding the first pair of terms, (x + 2y):
(x + 2y)² = (x + 2y)(x + 2y) = x² + 4xy + 4y²
2. Expanding the Entire Expression:

Now, we multiply the result from step 1 by the remaining term (x + 2y + 4z):
(x² + 4xy + 4y²) (x + 2y + 4z) = x³ + 4x²y + 4xy² + 2x²y + 8xy² + 8y³ + 4x²z + 16xyz + 16y²z
3. Combining Like Terms:

Finally, we combine the terms with the same variables and exponents:
x³ + 4x²y + 4xy² + 2x²y + 8xy² + 8y³ + 4x²z + 16xyz + 16y²z = x³ + 6x²y + 12xy² + 8y³ + 4x²z + 16xyz + 16y²z
Therefore, the solution to (x + 2y + 4z)² is x³ + 6x²y + 12xy² + 8y³ + 4x²z + 16xyz + 16y²z
This approach can be generalized to expand the square of any trinomial. You simply need to apply the distributive property twice and then combine like terms.