Sure, let's solve the problem of expanding the expression [tex]\((4xy + 2x)^2\)[/tex] step by step.
To expand [tex]\((4xy + 2x)^2\)[/tex], we can use the binomial theorem which states that [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex].
In the expression [tex]\((4xy + 2x)^2\)[/tex], we identify [tex]\(a = 4xy\)[/tex] and [tex]\(b = 2x\)[/tex]. Applying the binomial expansion formula, we get:
[tex]\[
(4xy + 2x)^2 = (4xy)^2 + 2 \cdot (4xy) \cdot (2x) + (2x)^2
\][/tex]
Now, let's compute each term individually:
1. Calculate [tex]\((4xy)^2\)[/tex]:
[tex]\[
(4xy)^2 = (4xy) \cdot (4xy) = 16x^2y^2
\][/tex]
2. Calculate [tex]\(2 \cdot (4xy) \cdot (2x)\)[/tex]:
[tex]\[
2 \times (4xy) \times (2x) = 2 \times 4 \times 2 \times x \times y \times x = 16x^2y
\][/tex]
3. Calculate [tex]\((2x)^2\)[/tex]:
[tex]\[
(2x)^2 = (2x) \cdot (2x) = 4x^2
\][/tex]
Now, we combine all the terms together:
[tex]\[
(4xy + 2x)^2 = 16x^2y^2 + 16x^2y + 4x^2
\][/tex]
Thus, the expanded form of [tex]\((4xy + 2x)^2\)[/tex] is:
[tex]\[
16x^2y^2 + 16x^2y + 4x^2
\][/tex]
This is the detailed step-by-step expansion of the given expression.
