Factoring a quadratic polynomial (absolute beginner level), are both answers correct? I'm following video tutorials on factoring quadratic polynomials. So I'm given the polynomial:
$$x^2 + 3x - 10$$
And I'm given the task of finding the values of $a$ and $b$ in:
$$(x + a) (x + b)$$
Obviously the answer is:
$$(x + 5)(x - 2)$$
However the answer can be also:
$$(x - 2) (x + 5)$$
I just want to make sure if the question asks for the values of '$a$' and '$b$', then '$a$' can be either $5$ or $-2$, and '$b$' can be either $5$ or $-2$.
Therefore if a question asks what are the values of '$a$' and '$b$' both the following answers are correct:
Answer $1$
$a = -2$
$b = 5$
or
Answer $2$
$a = 5$
$b = -2$
I'm sure this is a completely obvious question, but I'm just a beginner in this. 
 A: For commutative property of product we have that
$$(x + 5)(x - 2)=(x - 2)(x + 5)$$
note that also
$$(-x + 2)(-x - 5)$$
is a correct factorization.
A: .Yes, you are correct. Since $(x+5)(x-2) = (x-2)(x+5)  = x^2 + 3x-10$, we note that $a$ and $b$ may either take the values $(5,-2)$ or $(-2,5)$. 

I would consider providing just one of the two solutions to be insufficient, since the question itself ask for the values of $a$ and $b$, but nowhere mentions that they are unique. However, any question saying "find the values of $a$ and $b$" is wrong with the word "the" : they are assuming uniqueness of $a$ and $b$, which is not the case.The question as quoted by you includes the word "the" , and this is misleading.
A: You are right.
$$(x+a)(x+b)=x^2+(a+b)x+ab$$
and by identification with $x^2+3x-10$,
$$\begin{cases}a+b=3,\\ab=-10.\end{cases}$$
This is a non-linear system of equations, and given commutativity of addition and multiplication, it is clear that if $(u,v)$ is a solution, so is $(v,u)$. 

Now one may wonder if more than two solutions could exist. As $a=0$ cannot be a solution, we can write
$$3a=(a+b)a=a^2+ab=a^2-10$$
which is the original equation (with a sign reversal)
$$a^2-3a-10=0.$$
To be able to conclude, you must invoke the fundamental theorem of algebra, which implies that a quadratic equation cannot have more than two roots.
So there are exactly these two solutions: $a=-2,b=5$ and $a=5,b=-2$.
