Explanation:
since the polynomial is in the form
+x² - nx - m
we know that we are looking for 2 factors of x (to create x²).
and they must be in the form of
(x + a)(x - b)
to create the negative "m" at the end.
and "b" must be larger than "a" to create the negative "n" in the middle.
let's do the multiplication and compare the parts with our actual expression :
(x + a)(x - b) = x² - bx + ax - ab = x² + (a - b)x - ab
so, we know now in numbers that
-ab = -44
ab = 44
and
a - b = -7
a = b - 7
the short solution here is trial and error with the factors of 44 :
44 = 1×44 = 2×22 = 4×11
ah, only one combination has a difference of the factors of 7 or -7 : 4 and 11.
since b has to be larger (as a - b = -7), we have a = 4 and b = 11
and the factorization of our original expression is
f(x) = (x + 4)(x - 11)
FYI
if there is not such a short list of options to try, we can also go the full formal way :
we can use the second equation as identity in the first equation and get
(b - 7)b = 44
b² - 7b = 44
b² - 7b - 44 = 0
by the way, it is not surprising that it looks the same way as the original expression in x, because the factorization uses exactly the zero solutions for the original expression.
anyway, the formal solution to a quadratic equation of
ax² + bx + c = 0 is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
x = b (the variable of our equation above)
a = 1
b = -7
c = -44
b = (7 ± sqrt((-7)² - 4×1×-44))/(2×1) =
= (7 ± sqrt(49 + 176))/2 = (7 ± sqrt(225))/2 =
= (7 ± 15)/2
b1 = (7 + 15) / 2 = 22/2 = 11
b2 = (7 - 15) / 2 = 8/2 = 4
based again on the original equations and the fact that b has to be the larger number, we get again (of course) that b1 is the valid solution and
b = 11 and a = 4