One of the factors of $\backslash (6x^2-24\; xy+17x+24y^2-34y+5\backslash )$ is :

Given:

The expression to factorize is 6x² - 24xy + 17x + 24y² - 34y + 5

Calculation:

Let's try to group terms or use a systematic approach.

Consider the quadratic terms: 6x² - 24xy + 24y²

We can factor out 6: 6(x² - 4xy + 4y²)

Recognize the perfect square trinomial: 6(x - 2y)²

So the expression becomes: 6(x - 2y)² + 17x - 34y + 5

Notice that 17x - 34y can be factored as 17(x - 2y).

Let P = (x - 2y).

The expression now becomes: 6P² + 17P + 5

This is a quadratic expression in P. We can factor this quadratic.

We need two numbers that multiply to 6 × 5 = 30 and add up to 17. These numbers are 15 and 2.

6P² + 15P + 2P + 5

(6P² + 15P) + (2P + 5)

3P(2P + 5) + 1(2P + 5)

(3P + 1)(2P + 5)

Now substitute P = (x - 2y) back into the factored expression:

[3(x - 2y) + 1][2(x - 2y) + 5]

[3x - 6y + 1][2x - 4y + 5]

So, the factors are (3x - 6y + 1) and (2x - 4y + 5).

∴ The correct answer is option 4.