Q9 Add the polynomial expressions 3y-4y^2+7y+2 and 5y^2+6-8y.

To add the polynomial expressions \( 3y - 4y^2 + 7y + 2 \) and \( 5y^2 + 6 - 8y \), we first combine like terms.

1. **Identify and arrange the terms:**

- From the first polynomial:
- \( -4y^2 \) (the \( y^2 \) term)
- \( 3y + 7y = 10y \) (the \( y \) terms)
- \( 2 \) (the constant term)

- From the second polynomial:
- \( 5y^2 \) (the \( y^2 \) term)
- \( -8y \) (the \( y \) term)
- \( 6 \) (the constant term)

2. **Add the like terms:**

- For \( y^2 \) terms:
\[
-4y^2 + 5y^2 = 1y^2 \text{ or simply } y^2
\]

- For \( y \) terms:
\[
10y - 8y = 2y
\]

- For the constant terms:
\[
2 + 6 = 8
\]

3. **Combine all the results:**
The sum of the polynomials is:
\[
y^2 + 2y + 8
\]

Thus, the final result is:
\[
\boxed{y^2 + 2y + 8}
\]

Q10 Subtract.

(7x^2+11x+9)-(3x^2+5x+4)

To subtract the polynomial expressions \( (7x^2 + 11x + 9) - (3x^2 + 5x + 4) \), we will follow these steps:

1. **Distribute the negative sign** to the second polynomial:
\[
7x^2 + 11x + 9 - 3x^2 - 5x - 4
\]

2. **Combine like terms**:

- For \( x^2 \) terms:
\[
7x^2 - 3x^2 = 4x^2
\]

- For \( x \) terms:
\[
11x - 5x = 6x
\]

- For constant terms:
\[
9 - 4 = 5
\]

3. **Combine all the results**:
The result of the subtraction is:
\[
4x^2 + 6x + 5
\]

Thus, the final answer is:
\[
\boxed{4x^2 + 6x + 5}
\]

Q11 Multiply the polynomials (2x-2)(3x+5)

To multiply the polynomials \( (2x - 2)(3x + 5) \), we can use the distributive property (also known as the FOIL method for binomials), which stands for First, Outer, Inner, Last.

1. **Multiply the First terms:**
\[
2x \cdot 3x = 6x^2
\]

2. **Multiply the Outer terms:**
\[
2x \cdot 5 = 10x
\]

3. **Multiply the Inner terms:**
\[
-2 \cdot 3x = -6x
\]

4. **Multiply the Last terms:**
\[
-2 \cdot 5 = -10
\]

5. **Combine all the results:**
\[
6x^2 + 10x - 6x - 10
\]

6. **Combine like terms:**
\[
6x^2 + (10x - 6x) - 10 = 6x^2 + 4x - 10
\]

Thus, the final result of the multiplication is:
\[
\boxed{6x^2 + 4x - 10}
\]

Q12 Which of the following responses shows that polynomials form a closed system under multiplication?

A. x^2+3x/x-1
B. (2x^2+3x)(x-4)
1x-3/ . 5

To determine which response shows that polynomials form a closed system under multiplication, we need to analyze each option and see if the result is a polynomial.

**A. \( \frac{x^2 + 3x}{x - 1} \)**

This expression represents a rational function, as it is the division of one polynomial (\(x^2 + 3x\)) by another (\(x - 1\)). The result is not a polynomial because polynomials cannot have variables in the denominator. So, this option does not demonstrate closure under multiplication.

**B. \( (2x^2 + 3x)(x - 4) \)**

This expression shows the multiplication of two polynomials:

- \( 2x^2 + 3x \) is a polynomial.
- \( x - 4 \) is also a polynomial.

When we multiply these two polynomials, the result will also be a polynomial.

Let's perform the multiplication:

\[
(2x^2 + 3x)(x - 4) = 2x^2 \cdot x + 2x^2 \cdot (-4) + 3x \cdot x + 3x \cdot (-4)
\]
\[
= 2x^3 - 8x^2 + 3x^2 - 12x
\]
\[
= 2x^3 - 5x^2 - 12x
\]

The result \( 2x^3 - 5x^2 - 12x \) is indeed a polynomial.

Thus, the correct answer that demonstrates closure under multiplication of polynomials is:

**B. \( (2x^2 + 3x)(x - 4) \)**

Therefore, we can conclude:
\[
\boxed{B}
\]

Q13 Divide 20x^2+15x+10 by -5x

To divide the polynomial \( 20x^2 + 15x + 10 \) by \( -5x \), we will divide each term of the polynomial by \( -5x \).

1. **Divide the first term:**
\[
\frac{20x^2}{-5x} = -4x
\]

2. **Divide the second term:**
\[
\frac{15x}{-5x} = -3
\]

3. **Divide the third term:**
\[
\frac{10}{-5x} = -\frac{2}{x}
\]

Putting it all together, the result of the division is:

\[
-4x - 3 - \frac{2}{x}
\]

Therefore, the final result is:

\[
\boxed{-4x - 3 - \frac{2}{x}}
\]

Q14 The design for a rectangular solid shipping container has a volume of 30x^3+5x^2-15x where x represents inches. The box has a height of 5x. Find the container’s cross-sectional area using polynomial division.

To find the cross-sectional area of the rectangular solid shipping container, we need to divide the volume polynomial \( 30x^3 + 5x^2 - 15x \) by the height \( 5x \).

We will use polynomial long division for this process.

### Step 1: Set Up the Division

We are dividing \( 30x^3 + 5x^2 - 15x \) by \( 5x \).

### Step 2: Perform the Division

1. **Divide the leading term:**
\[
\frac{30x^3}{5x} = 6x^2
\]
Write down \( 6x^2 \).

2. **Multiply \( 6x^2 \) by \( 5x \):**
\[
6x^2 \cdot 5x = 30x^3
\]

3. **Subtract this product from the original polynomial:**
\[
(30x^3 + 5x^2 - 15x) - (30x^3) = 5x^2 - 15x
\]

4. **Bring down the next term (if there are more terms); they are already included here.**

5. **Repeat the division:**
\[
\frac{5x^2}{5x} = x
\]
Write down \( x \).

6. **Multiply \( x \) by \( 5x \):**
\[
x \cdot 5x = 5x^2
\]

7. **Subtract this product from the remaining polynomial:**
\[
(5x^2 - 15x) - (5x^2) = -15x
\]

8. **Repeat the division:**
\[
\frac{-15x}{5x} = -3
\]
Write down \( -3 \).

9. **Multiply \( -3 \) by \( 5x \):**
\[
-3 \cdot 5x = -15x
\]

10. **Subtract this product from the remaining polynomial:**
\[
(-15x) - (-15x) = 0
\]

Since there are no more terms to bring down, we have completed our polynomial division.

### Conclusion

The result of the division is:
\[
6x^2 + x - 3
\]

Thus, the container’s cross-sectional area is given by:
\[
\boxed{6x^2 + x - 3}
\]