Question

1. $8c^2 - c + 3 + 2c^2 - c + 2$
2. $6a^2b^2 + 3ab^2 - a^2b^2 - 4ab^2$
3. write the perimeter of the rectangle as a simplified expression.

rectangle with length $7x + 1$ and width $x - 3$

Explanation:

Problem 15: Simplify \( 8c^2 - c + 3 + 2c^2 - c + 2 \)

Step1: Combine like terms for \( c^2 \)

\( 8c^2 + 2c^2 = (8 + 2)c^2 = 10c^2 \)

Step2: Combine like terms for \( c \)

\( -c - c = (-1 - 1)c = -2c \)

Step3: Combine constant terms

\( 3 + 2 = 5 \)

Step4: Combine all simplified terms

\( 10c^2 - 2c + 5 \)

Step1: Combine like terms for \( a^2b^2 \)

\( 6a^2b^2 - a^2b^2 = (6 - 1)a^2b^2 = 5a^2b^2 \)

Step2: Combine like terms for \( ab^2 \)

\( 3ab^2 - 4ab^2 = (3 - 4)ab^2 = -ab^2 \)

Step3: Combine all simplified terms

\( 5a^2b^2 - ab^2 \)

Step1: Substitute length and width into the perimeter formula

\( P = 2[(7x + 1) + (x - 3)] \)

Step2: Simplify inside the parentheses

\( (7x + 1) + (x - 3) = 7x + x + 1 - 3 = 8x - 2 \)

Step3: Multiply by 2

\( P = 2(8x - 2) = 16x - 4 \)

Answer:

\( 10c^2 - 2c + 5 \)

Problem 17: Simplify \( 6a^2b^2 + 3ab^2 - a^2b^2 - 4ab^2 \)