Break the constant term '24' into ''25-1'' (both of which are perfect squares, viz, 25 = 5^2 and 1= 1^2) , and then complete the squares.
289x2+ 170x + 24 - 38y - 361y2
289x2+ 170x + (25 - 1) - 38y - 361y2
289x2+ 170x + 25 - 1 - 38y - 361y2
(289x2+ 170x + 25) - (1 + 38y + 361y2)
{(17x)2 + 2. 17x. 5 + (5)2 } - {12 + 2. 1. 19y + (19y)2}
(17x + 5)2 - (1 + 19y)2
Now as you know a^2 - b^2 = (a+b) (a-b),
(17x + 19y + 6) (17x - 19y + 4) is the required solution.
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