<s>[INST] Factorise $y^2 -3yz -10z^2$ How do I solve this question? I have looked at the problem several times. However, I cannot find a viable solution. I believe that it is a perfect square trinomial problem. 
 [/INST] I made up a method with no guesswork at all, Factoring Quadratics: Asterisk Method 
and my earlier answer at How to factor the quadratic polynomial $2x^2-5xy-y^2$? 
We require that $b^2 - 4ac$ be a perfect square, and (positive) $\delta = \sqrt {b^2 - 4ac},$ otherwise factoring is impossible anyway. Then: we may simply take $$ a_1 = \gcd \left( a, 
  \frac{(b + \delta)}{2}  \right) ; \; \; \; \; \; a_2 = \frac{a}{a_1}  $$ without paying attention to any prime factorizations and
$$  a x^2 + b x y + c y^2 =  \; \left(a_2x+ \left( \frac{b + \delta}{2a_1} \right) y \right) \; \; \left(a_1x+ \left( \frac{b - \delta}{2a_2} \right) y \right) \;    $$ in integers.
You have $a=1, b= -3, c=-10, b^2 - 4 a c = 49, \delta = 7.$ So $a_1 = 1, a_2 = 1.$ Then $(b+\delta)/ (2 a_1) = 4/2 = 2,$ and $(b-\delta)/ (2 a_2) = -10/2 = -5,$ so $x^2 - 3 x y - 10 y^2 = (x + 2 y)(x - 5 y).$
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