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I am a self-taught software engineer who is finally getting back into math. So this is a basic question.

The following expression:

$-4(3w-u)-6(-4u-4w)$ confuses me. As I do not know why I should not solve it by SUBTRACTING $-4(3w-u)$ from the result of $6(-4u-4w)$. However, the question was answered as if the minus sign was ignored and instead became :

$$-4(3w-u)+ (-6(-4u-4w))$$

What am I not understanding. Thanks for your help.

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Recall that for any numbers $a, b$, $$a - b = a + (-b)$$ and so, for example $$-a + b = b + -a = b - a\;\text{ and}\;-a - b = -a + (-b)$$

Put differently, subtracting $b$ from $a$ is the same as adding $-b$ to $a$.

Now, looking at your problem: subtracting $6(-4u - 4w)$ is the same as adding $-6(-4u - 4w)$, and similarly, adding $-4(3w - u)$ to $-6(-4u - 4w)$ is the same as subtracting $4(3w - u)$ from $-6(-4u - 4w)$.

$$-4(3w-u)\color{blue}{\bf -6}(-4u-4w) = -4(3w-u) \color{blue}{\bf +( -6}(-4u - w))\tag{given}$$ Distributing gives us:

$$ \left[-4\cdot 3 w + -4(-u)\right] + \left[(-6)\cdot (-4u) + (-6)(-4w)\right]$$

$$ = -12w + 4u + 24u + 24w\quad\quad\tag{simplify}$$ $$ = 24w + -12w + 4 u + 24 u \quad\quad\tag{rearrange to gather like terms}$$ $$= (24 + -12)w + (4 + 24)u \quad\quad\tag{"reverse" distribution}$$ $$ = 12w + 28u\quad\quad\tag{simplify}$$

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nice Amy nicely done. ;-) – Babak S. Mar 21 '13 at 7:04

Subtracting some number $x$ is the same as adding $-x$. So you could think of it as subtracting $-4(3w - u)$ from $- 6(-4u - 4w)$ (don't forget the minus sign), but there is no good reason why you should. We can simplify the original expression using distributivity:

\begin{align}-4(3w - u) -6(-4u - 4w) &= (-12w + 4u) - (-24u - 24w) \\ &= -12w + 4u + 24u + 24w \\ &= 12w + 28u\end{align}

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