rule applied when negating a polynomial expression I'm wondering what mathematical rule is applied when negating a polynomial expression. For example, in high school it is taught that $-(-6x^2 + 15x - 5) = 6x^2 -15x + 5$, but what rule(s) is applied here? Some say it's the distributive property (distributing the negation), but I'm not sure about that because negation is an operation, not a number.
It's easier for me to see that $-(-6) = 6$ (for example) because of the rule that the negation of a negative number is the corresponding positive number.
 A: Negation is an operation, yes. Specifically, it is multiplication by $-1$. That's where distributivity comes into play.
To see why this is true, note that $$x+-x=0=0\cdot x=1\cdot x+(-1)\cdot x=x+(-1)\cdot x,$$ so $$-x=(-1)\cdot x.$$
A: The distributive property is used to distribute the factor of $-1$ over all the terms:
$$
\begin{align}
-1\cdot(-6x^2+15x-5)
&=-1(-6x^2)+-1(15x)+-1(-5)\\
&=6x^2-15x+5
\end{align}
$$
A: It's the distributive property of multiplication over addition: $$-(-6x^2 + 15x - 5) = -1\cdot(6x^2 + 15x - 5) = -1\cdot 6x^2 + -1\cdot 15x - (-1)\cdot 5 = -6x^2 -15x + 5$$
A: $(6x^2-15x+5)+(-6x^2+15x-5)=0$, now add the negative of $-6x^2+15x-5$ to each side.
A: That $\rm\ {-}(-6x^2\! + 15x - 5)\, =\, 6x^2\! -15x + 5\ $ does not require the distributive law. Rather, it is true more generally in any abelian (commutative) group that $\rm\:-(a+b) = -a + -b,\:$ because $$\rm\:-a + -b + a + b\, =\, -a + a + -b + b\, =\, 0 + 0\, =\, 0$$
In the special case of a ring, one has that $\rm\: -x = (-1)x,\:$ so one can instead use distributivity $$\rm -(a+b)\, =\, (-1)(a+b)\, =\, (-1)a + (-1)b\, =\, -a + -b$$
which then yields the Law of Signs $\rm\,\ (-a)(-b) = ab\,\ $ and related properties.
But, conceptually, your inference is purely a property of groups, true also in nonabelian groups 
$$\rm (abc)^{-1} =\, c^{-1}b^{-1}a^{-1}$$
A: Suppose we know the rule that $(-a) = (-1) a$ and we know the distributive property for multiplication over addition. Then,
$$-(a+b) = (-1) (a+b) = (-1)a + (-1) b = (-a) + (-b)$$
so we've discovered a new rule: negation distributes over addition. In some form it distributes over subtraction too:
$$-(a-b) = (-1) (a-b) = (-1) - (-1) b = (-a) - (-b) = (-a) + b$$
or across many terms
$$-(a+b-c+d-e) = (-1) (a+b-c+d-e) = \cdots = -a - b + c -d + e $$
Distribution of negation fits into the same spatial reasoning that distribution of multiplication does; visually, one "sees" the same thing happening in both cases. For the vast majority of purposes, it is not useful to distinguish between the two.
