Verified answer

To factorize the expression 2x^2 + y^2 + 8z^2 - 2 √2xy + 4 √2yz - 8xy, we can group the terms:

(2x^2+ y^2) + (8z^2 + 4√2yz) - (2√2xy + 8xy)

Now let's factor out common terms from each group:

2x^2 + y^2 = (x√2 + y)(x√2 - y)

8z^2 + 4√2yz = 4z(2z + √2y)

-2√2xy + 8xy = -2xy(√2 - 4)

Therefore, the factored form of the expression 2x^2 + y^2 + 8z^2 - 2√2xy + 4√2yz - 8xy is:

(x√2 + y)(x√2 - y) + 4z(2z + √2y) - 2xy(√2 - 4)

[tex]5 x - 3y - 4 = 0 \\ \\ 10x - 6y - 9 = 0[/tex]

[tex]Multiply \: by \: 2 \\ \\ 2(5x - 3y - 4) = 0 \\ \\ 10x - 6y - 8 = 0[/tex]

[tex]10x - 6y - 8 = 0 \\ \\ 10x - 6y - 9 = 0[/tex]

[tex]Distance \: between \: the \\ parallel \: lines[/tex]

[tex] = \frac{ - 8 + 9}{ \sqrt{100 + 36} } \\ \\ = \frac{1}{ \sqrt{136} } \\ \\ = \frac{1}{2 \sqrt{34} } [/tex]