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Chapter 0: Problem 81

Find each product. $$(2 x+y+1)^{2}$$

Short Answer

Expert verified
The product of \( (2x + y + 1)^{2} \) is \( 4x^{2} + 4xy + 2x + y^{2} + 2y + 1\).

Step by step solution

01

Rewrite the expression

Rewrite the expression \( (2x + y + 1)^{2} \) as \( (2x + y + 1) * (2x + y + 1) \) to clearly show that it's a product of two identical expressions.
02

Expand using the binomial theorem

The binomial theorem shows the multiplication of two identical binomials. We can apply this theorem here: \( (2x + y + 1)*(2x + y + 1) \). The result is \( (2x)* (2x) + 2 * (2x)* (y) + 2 * (2x)* (1) + (y)* (y) + 2 * (y)* (1) + (1)* (1) \).
03

Simplification

Now simplify the result from step 2. It will be \( 4x^{2} + 4xy + 2x + y^{2} + 2y + 1\).

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