Math help please help me please ??

If the coefficients a,b,c are integers of small or moderate size, it is often possible to factor "by inspection." An example is 3x^2 - 4x + 1, where experience will suggest (3x - 1)(x - 1), which turns out to be 3x^2 - 4x + 1. It's always worth seeking such a solution if "a" and "c" are primes or small integers with few factors.

A second method is "completing the square." Here one seeks a perfect square whose first two terms are ax^2 + bx. The example above is not a good candidate, since 3 itself is not a perfect square. However, something like 9x^2 + 12x - 45 could be regarded as (3x+2)^2 - 49, so the factors are (3x+2)+7 and (3x+2)-7; which should be rewritten as (3x+9)(3x-5) = 3*(x+3)(3x-5). Of course the same result could have been achieved by writing 3*(3x^2 + 4x - 15) and then trying "inspection" to get 3*(3x-5)(x+3).

When these methods fail, the quadratic formula can be used to find factors that may or may not be rational. In the case of 9x^2+12x-45, if you don't just see the factors right away, you can set
9x^2 + 12x - 45 = 0 and then use
x = (-b +/- sqrt(b^2-4ac)/2a
= (-12 +/- sqrt(144+1620))/18
= -12/18 +/- (1/18)*sqrt(1764)
= -12/18 +/- 42/18 = 30/18 or -54/18 = 5/3 or-3.
This shows that (x+3) and (3x-5) are factors. Their product is 3x^2 + 4x - 15, which must be a constant multiple of the original expression. Evidently the constant multiple is 1/3, so the full factorization of 9x^2 + 12x - 45 is 3(x+3)(3x-5), as before.

Next consider x^2 + 14x - 6. If you set
x^2 + 14x - 6 = 0, you get
x = -7 +/- (1/2)*sqrt(196+24)
= -7 +/- sqrt(55), so the only linear factors are
(x + 7 + sqrt(55)) and (x + 7 - sqrt(55)).
As the operation of "factoring" is usually a search for rational factors, you could also report that
x^2 + 14x - 6 is "unfactorable," or "has no rational factors."
T