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ить на множители многочлен

(1+4x^(2))y^(2)+2(2x-y)(1+2xy)+1

ить на множители многочлен $\newline$ $\left(1+4 x^{2}\right) y^{2}+2(2 x-y)(1+2 x y)+1$

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Factor sums and differences of cubes

Full solution

Q. ить на множители многочлен

\newline

\left(1+4 x^{2}\right) y^{2}+2(2 x-y)(1+2 x y)+1

Recognize pattern: Recognize the pattern in the given polynomial; it resembles the expansion of $(a+b)^2$ , which is $a^2 + 2ab + b^2$ .
Identify $a$ and $b$ : Identify $a$ and $b$ in the polynomial. Here, $a = y$ and $b = (1+2xy)$ .
Check with $(a+b)^2$ : Write down the square of $(a+b)$ to check if it matches the given polynomial: $(y + (1+2xy))^2 = y^2 + 2y(1+2xy) + (1+2xy)^2$ .
Expand $(1+2xy)^2$ : Expand $(1+2xy)^2$ to see if it matches the given polynomial: $(1+2xy)^2 = 1^2 + 2*(1*2xy) + (2xy)^2 = 1 + 4xy + 4x^2y^2$ .
Match expanded forms: Now, check if the expanded form of $(y + (1+2xy))^2$ matches the given polynomial: $y^2 + 2y(1+2xy) + (1+4x^2)y^2 = y^2 + 2y + 4xy^2 + 1 + 4x^2y^2$ .

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