To simplify, remove parentheses and combine like terms:

$\displaystyle (12p^2 +3p -12) + (-3p^2 - p +10)$

$\displaystyle 12p^2 -3p^2 +3p - p -12 +10$

$\displaystyle 9p^2 +2p-2$

To simplify, remove parentheses and combine like terms:

$\displaystyle q^2 + 4q + 8 + 2q^2 -2q - 8$

$\displaystyle q^2 + 2q^2 +4q - 2q +8 - 8$

$\displaystyle 3q^2 +2q$

Note that adding or subtracting a zero to the end of this equation is unnecessary. 

To simplify, simply remove the parentheses and combine like terms:

$\displaystyle (4y^2 +2y +6) + (y^2 +3y -10)$

$\displaystyle 4y^2 +y^2 +2y +3y +6 -10$

$\displaystyle 5y^2 +5y - 4$

We can represent this as a subtraction of trinomials.  

  (12F + 6D + 2G) – (3F + 2D + 1G) = 9F + 4D + 1G. 

1. Represent the situation with two sets of trinomials:

Before your friends ate the fruit:

$\displaystyle 7A+ 12B+16C$

The fruit your friends ate:

$\displaystyle 3A+9B+4C$

2. Subtract the first trinomial from the second trinomial:

$\displaystyle (7A+12B+16C)-(3A+9B+4C)=$

$\displaystyle 7A-3A+12B-9B+16C-4C=4A+3B+12C$

To solve this expression, merely remove the parentheses (bearing in mind that because the second trinomial is being subtracted, it will be negative) and combine like terms:

$\displaystyle (2n^ 2 - 4n + 16) - (3n^2 + 4n -12)$

$\displaystyle 2n^ 2 -4n + 16 -3n^2 -4n +12$

$\displaystyle 2n^2 -3n^2 -4n -4n +16 +12$

$\displaystyle -n^2 -8n +28$

To simplify, remove parentheses and combine like terms:

$\displaystyle 5r^2 +3r - 5 -4r^2 -5r +8$

$\displaystyle 5r^2 -4r^2 +3r-5r-5+8$

$\displaystyle r^2 -2r +3$

To simplify, remove parentheses and combine like terms:

$\displaystyle m^2 +2m +2 -3m^2 +2m -4$

$\displaystyle m^2 -3m^2 +2m +2m +2 -4$

$\displaystyle -2m^2 +4m -2$

$\displaystyle \frac{x^{2}-6x-7}{x^{2}-2x-3}=\frac{(x+1)(x-7)}{(x+1)(x-3)}$

Once simplified, (x+1) appears on both the numerator and denominator, meaning we can cancel out both of them.

Which gives us:

$\displaystyle \frac{x-7}{x-3}$

$\displaystyle x^4-2x^3+7 \div x^2+4x+4$ can be divided using long division. 

The set up would look very similar to the division of real numbers, such as when we want to divide 10 by 2 and the answer is 5.

The first step after setting up the "division house" is to see what the first term in the outer trinomial needs to multiplied by to match the $\displaystyle x^4$ in the house. In this case, it's $\displaystyle x^2$. $\displaystyle x^2$ will be multiplied across the other two terms in the outer trinomial and the product will be subtracted from the expression inside the division house. The following steps will take place in the same way. 

While we could continue to divide, it would require the use of fraction exponents that would make the answer more complicated. Therefore, the term in red will be the remainder. Because this remainder is still subject to be being divided by the trinomial outside of the division house, we will make the remainder part of the final answer by writing it in fraction form:

$\displaystyle \frac{-56x-73}{x^2+4x+4}$

Therefore the final answer is

$\displaystyle x^2-6x+20+\frac{-56x-73}{x^2+4x+4}$