1. Subtract 2a(a + b - c) - 2c(a + b - c) from 2c(-a + b + c). 2. A bus travels 40 km in 30 minutes. If the speed of the bus remains the same, how far can it travel in 3 hours?

Steps to Solve

1. Simplify the first expression

We need to simplify $2a(a + b - c) - 2c(a + b - c)$ by factoring.

$2a(a + b - c) - 2c(a + b - c) = (2a - 2c)(a + b - c)$

2. Expand and simplify the first expression further

Expand $(2a - 2c)(a + b - c)$:

$(2a - 2c)(a + b - c) = 2a^2 + 2ab - 2ac - 2ac - 2bc + 2c^2 = 2a^2 + 2ab - 4ac - 2bc + 2c^2$

3. Rewrite the subtraction problem

We are asked to subtract the simplified expression from $2c(-a + b + c)$. So we have:

$2c(-a + b + c) - (2a^2 + 2ab - 4ac - 2bc + 2c^2)$

4. Expand the first term of the subtraction

Expand $2c(-a + b + c)$:

$2c(-a + b + c) = -2ac + 2bc + 2c^2$

5. Perform the subtraction

Now we subtract the expressions we derived in steps 2 and 4:

$(-2ac + 2bc + 2c^2) - (2a^2 + 2ab - 4ac - 2bc + 2c^2) = -2ac + 2bc + 2c^2 - 2a^2 - 2ab + 4ac + 2bc - 2c^2$

6. Simplify the resulting expression

Combine like terms: $-2a^2 - 2ab + (-2ac + 4ac) + (2bc + 2bc) + (2c^2 - 2c^2) = -2a^2 - 2ab + 2ac + 4bc$

Therefore the simplified expression is $-2a^2 - 2ab + 2ac + 4bc$.

7. Calculate the speed of the bus

The bus travels 40 km in 30 minutes, which is 0.5 hours. We can find the speed using the formula:

$speed = \frac{distance}{time}$

$speed = \frac{40 \text{ km}}{0.5 \text{ hours}} = 80 \text{ km/h}$

8. Calculate the distance traveled in 3 hours

Now, we need to find the distance the bus travels in 3 hours at the same speed. We use the formula:

$distance = speed \times time$

$distance = 80 \text{ km/h} \times 3 \text{ hours} = 240 \text{ km}$