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gmany
[tex]4x^2+y^2+4xy+8x+4y+4\\\\=\underbrace{(2x)^2+2\cdot2x\cdot y+y^2}_{(a+b)^2=a^2+2ab+b^2}+8x+4y+4\\\\=\underbrace{(2x+y)^2+2(2x+y)\cdot2+2^2}_{(a+b)^2=a^2+2ab+b^2}\\\\=[(2x+y)+2]^2=(2x+y+2)^2[/tex]
MrRoyal

The factors of the polynomial are (2x + y + 2) and (2x + y + 2)

The expression is given as:

[tex]\mathbf{4x^2 + y^2 + 4xy + 8x + 4y + 4}[/tex]

Rewrite as:

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = (2x)^2 + 4xy + y^2 + 8x + 4y + 4}[/tex]

Expand

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = (2x)^2 + 2xy + 2xy + y^2 + 8x + 4y + 4}[/tex]

Factorize

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = 2x(2x + y) + y(2x + y) + 8x + 4y + 4}[/tex]

Factor out 2x + y

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = (2x + y)(2x + y) + 8x + 4y + 4}[/tex]

Express as squares

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = (2x + y)^2 + 8x + 4y + 4}[/tex]

Factorize 8x + 4y

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = (2x + y)^2 + 4(2x + y) + 4}[/tex]

Let z = 2x + y

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = z^2 + 4z + 4}[/tex]

Expand

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = z^2 + 2z + 2z + 4}[/tex]

Factorize

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = z(z + 2) + 2(z + 2)}[/tex]

Factor out z + 2

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = (z + 2)(z + 2)}[/tex]

Express as squares

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = (z + 2)^2}[/tex]

Recall that: z = 2x + y

So, we have:

[tex]\mathbf{4x^2 + y^2 + 4xy + 8y + 4y + 4 = (2x + y + 2)^2}[/tex]

Hence, the factors of the polynomial are (2x + y + 2) and (2x + y + 2)

Read more about polynomial factors at:

https://brainly.com/question/12787576

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