What is the missing term? (10x - 4x^2) - (7x + ?) = 3x - 6x^2

1. Identify the equation: The problem is an algebraic equation where a missing term needs to be found. The equation is given as: $$(10x - 4x^2) - (7x + ?) = 3x - 6x^2$$
2. Simplify the left side of the equation: First, distribute the negative sign to the terms inside the second parenthesis. This changes the signs of each term within that parenthesis.
$$10x - 4x^2 - 7x - ? = 3x - 6x^2$$
3. Combine like terms on the left side: Combine the terms with 'x' and the terms with 'x^2'.
$$(10x - 7x) + (-4x^2) - ? = 3x - 6x^2$$
$$3x - 4x^2 - ? = 3x - 6x^2$$
4. Isolate the missing term: To find the missing term, we can rearrange the equation. Let the missing term be represented by 'y'. So, the equation becomes:
$$3x - 4x^2 - y = 3x - 6x^2$$
Now, isolate 'y' by moving other terms to the right side of the equation.
$$-y = 3x - 6x^2 - (3x - 4x^2)$$
5. Simplify the right side of the equation: Distribute the negative sign and combine like terms.
$$-y = 3x - 6x^2 - 3x + 4x^2$$
$$-y = (3x - 3x) + (-6x^2 + 4x^2)$$
$$-y = 0x + (-2x^2)$$
$$-y = -2x^2$$
6. Solve for the missing term: Multiply both sides by -1 to solve for 'y'.
$$y = 2x^2$$
However, looking at the original equation structure, the missing term is inside a parenthesis that is being subtracted. So, if the missing term is represented by '?', then the term being subtracted is $7x + ?$. Let's re-evaluate step 4 with this in mind.

Let the missing term be represented by 'M'. The equation is:
$$(10x - 4x^2) - (7x + M) = 3x - 6x^2$$
Expand the left side:
$$10x - 4x^2 - 7x - M = 3x - 6x^2$$
Combine like terms on the left:
$$3x - 4x^2 - M = 3x - 6x^2$$
Now, isolate '-M':
$$-M = 3x - 6x^2 - (3x - 4x^2)$$
$$-M = 3x - 6x^2 - 3x + 4x^2$$
$$-M = (3x - 3x) + (-6x^2 + 4x^2)$$
$$-M = -2x^2$$
Multiply by -1:
$$M = 2x^2$$
Let's check the options. The options are $-10x^2$, $-x^2$, $2x^2$, $-2x^2$. Our calculated value for M is $2x^2$.

Let's re-read the question carefully. It asks for the missing term. The structure is $(10x - 4x^2) - (7x + \text{missing term}) = 3x - 6x^2$.

Let the missing term be $T$. So, $$(10x - 4x^2) - (7x + T) = 3x - 6x^2$$
$$10x - 4x^2 - 7x - T = 3x - 6x^2$$
$$3x - 4x^2 - T = 3x - 6x^2$$
We want to find $T$.
Subtract $3x$ from both sides:
$$-4x^2 - T = -6x^2$$
Add $4x^2$ to both sides:
$$-T = -6x^2 + 4x^2$$
$$-T = -2x^2$$
Multiply by $-1$:
$$T = 2x^2$$

Let's consider the possibility that the missing term is not added to $7x$ but is a standalone term in the second parenthesis, i.e., $(7x \text{ missing term})$. The problem states $(7x + ?)$. This implies addition.

Let's re-examine the options and the problem. The problem is $(10x - 4x^2) - (7x + ?) = 3x - 6x^2$.
Let's denote the missing term by $X$.
$$(10x - 4x^2) - (7x + X) = 3x - 6x^2$$
$$10x - 4x^2 - 7x - X = 3x - 6x^2$$
$$3x - 4x^2 - X = 3x - 6x^2$$
Subtract $3x$ from both sides:
$$-4x^2 - X = -6x^2$$
Add $4x^2$ to both sides:
$$-X = -6x^2 + 4x^2$$
$$-X = -2x^2$$
Multiply by $-1$:
$$X = 2x^2$$

There might be a misunderstanding of the question or the options. Let's assume the question meant to ask for the term that should be subtracted from $(10x - 4x^2)$ to get $3x - 6x^2$. Let this term be $Y$.
$$(10x - 4x^2) - Y = 3x - 6x^2$$
$$Y = (10x - 4x^2) - (3x - 6x^2)$$
$$Y = 10x - 4x^2 - 3x + 6x^2$$
$$Y = (10x - 3x) + (-4x^2 + 6x^2)$$
$$Y = 7x + 2x^2$$
This does not match the structure of the second parenthesis $(7x + ?)$.

Let's go back to the original interpretation and check the options by substitution.

If the missing term is $-10x^2$: $$(10x - 4x^2) - (7x - 10x^2) = 10x - 4x^2 - 7x + 10x^2 = 3x + 6x^2 \neq 3x - 6x^2$$
If the missing term is $-x^2$: $$(10x - 4x^2) - (7x - x^2) = 10x - 4x^2 - 7x + x^2 = 3x - 3x^2 \neq 3x - 6x^2$$
If the missing term is $2x^2$: $$(10x - 4x^2) - (7x + 2x^2) = 10x - 4x^2 - 7x - 2x^2 = 3x - 6x^2$$
This matches the right side of the equation. So the missing term is $2x^2$.

Let's re-examine the provided options and the problem statement. The problem is: $(10x - 4x^2) - (7x + ?) = 3x - 6x^2$.
Let the missing term be $M$.
$$(10x - 4x^2) - (7x + M) = 3x - 6x^2$$
$$10x - 4x^2 - 7x - M = 3x - 6x^2$$
Combine like terms on the left side:
$$(10x - 7x) + (-4x^2) - M = 3x - 6x^2$$
$$3x - 4x^2 - M = 3x - 6x^2$$
To solve for $M$, we can isolate it. Subtract $3x$ from both sides:
$$-4x^2 - M = -6x^2$$
Add $4x^2$ to both sides:
$$-M = -6x^2 + 4x^2$$
$$-M = -2x^2$$
Multiply both sides by $-1$:
$$M = 2x^2$$

It appears there was a mistake in my initial interpretation of the options or the problem. Let's re-verify the substitution with the correct calculation.

If the missing term is $2x^2$, then the expression becomes:
$$(10x - 4x^2) - (7x + 2x^2)$$
$$= 10x - 4x^2 - 7x - 2x^2$$
Combine like terms:
$$= (10x - 7x) + (-4x^2 - 2x^2)$$
$$= 3x - 6x^2$$
This matches the right side of the given equation. Therefore, the missing term is $2x^2$.

Looking at the provided options, $2x^2$ is one of them.