To determine the factors of the expression 2xy + 8y + 8x + 32, we first need to rearrange and group the terms in a way that allows for factoring.
Step 1: Group like terms.
We can group the first two terms and the last two terms:
- (2xy + 8y) + (8x + 32)
Step 2: Factor out the common factors from each group.
From the first group, (2xy + 8y), we can factor out 2y:
- 2y(x + 4)
From the second group, (8x + 32), we can factor out 8:
- 8(x + 4)
Step 3: Rewrite the expression using the factored groups.
This gives us:
- 2y(x + 4) + 8(x + 4)
Step 4: Notice the common binomial factor.
We can see that both terms contain the common binomial factor (x + 4). Let’s factor that out:
- (x + 4)(2y + 8)
Step 5: Simplify further if possible.
In the term (2y + 8), we can factor out a 2:
- 2(y + 4)
Now the expression becomes:
- (x + 4)(2)(y + 4)
or simply:
- 2(x + 4)(y + 4)
Thus, the complete factorization of the given expression 2xy + 8y + 8x + 32 is:
- 2(x + 4)(y + 4)
Therefore, the factors of the expression are 2, (x + 4), and (y + 4).