How do I expand `(4x+2y)^2`?
To expand this expression, we need to use the FOIL method, considering that we are multiplying two binomials:
`(4x+2y)^2 = (4x+2y)(4x+2y)`
FOIL refers to doing the following process:
Product = First + Outer + Inner + Last
"What do these each mean?" you might ask. "First" refers to the first term inside the parenthesis. In our case, this means `4x*4x`. "Outer" means the product of the outer terms (`4x * 2y`). "Inner" means the product of the inner two terms (`2y*4x`). Finally, "Last" refers to the last terms (`2y*2y`).
In other words, all we are doing in these sorts of situations is multiplying each possible combination of one value from the left term and one value from the right term and adding all of those products together. What we get from the FOIL method is just a simplified way to find each possible combination. Our result for original expression is:
`(4x+2y)(4x+2y) = 4x*4x + 4x*2y+2y*4x + 2y*2y`
Now, we simplify each product:
`=16x^2 + 8xy + 8xy + 4y^2`
Now, we can combine like terms for our final result:
`= 16x^2 + 16xy + 4y^2`
You can actually expand a squared binomial in a special way:
`(a+b)^2 = a^2 + 2ab + b^2`
This method works with our problem. We can simply set a = 4x and b = 2y to get our same result:
`(4x+2y)^2 = (4x)^2 + 2(4x)(2y) + (2y)^2 = 16x^2 + 16xy + 4y^2`
I hope that helps!