Write the quadratic equation.

Find two numbers that multiply to -6 and add to -5. These numbers are -6 and 1.

The solutions to the quadratic equation are $\boxed{x=6 \text{ or } x=-1}$.

4a)

The scale is 1:500, meaning 1 cm on the model represents 500 cm in real life.
Model length = 5 cm.
Real-life length = $5 \text{ cm} \times 500 = 2500 \text{ cm}$.
Convert to meters: $2500 \text{ cm} = \frac{2500}{100} \text{ m} = 25 \text{ m}$.
The height of the room is $\boxed{25 \text{ m}}$.

The scale for length is $1:500$.
The scale for area is $(1:500)^2 = 1:500^2 = 1:250000$.
Real-life floor area = $36 \text{ m}^2$.
Convert real-life area to cm$^2$: $1 \text{ m}^2 = (100 \text{ cm})^2 = 10000 \text{ cm}^2$.
Real-life floor area = $36 \times 10000 \text{ cm}^2 = 360000 \text{ cm}^2$.
Area on model = Real-life area $\times \frac{1}{250000}$.
Area on model = $360000 \text{ cm}^2 \times \frac{1}{250000} = \frac{36}{25} \text{ cm}^2 = 1.44 \text{ cm}^2$.
The corresponding area on the floor of the model is $\boxed{1.44 \text{ cm}^2}$.

4b)

The scale for length is $1:500$.
The scale for volume is $(1:500)^3 = 1:500^3 = 1:125000000$.
Real-life volume = $120 \text{ m}^3$.
Convert real-life volume to cm$^3$: $1 \text{ m}^3 = (100 \text{ cm})^3 = 1000000 \text{ cm}^3$.
Real-life volume = $120 \times 1000000 \text{ cm}^3 = 120000000 \text{ cm}^3$.
Volume on model = Real-life volume $\times \frac{1}{125000000}$.
Volume on model = $120000000 \text{ cm}^3 \times \frac{1}{125000000} = \frac{120}{125} \text{ cm}^3$.

The corresponding volume of the model is $\boxed{0.96 \text{ cm}^3}$.

5)

Given that BA is parallel to CE, we can conclude that $\triangle CDE$ is similar to $\triangle BDA$.
This is because:

• $\angle DCE = \angle DBA$ (corresponding angles)
• $\angle CED = \angle BAD$ (corresponding angles)
• $\angle CDE = \angle BDA$ (common angle)

From the similarity $\triangle CDE \sim \triangle BDA$, we have:

We are given:
BA = 9 cm
CE = 4 cm
AE = 3 cm
Let DE = $x$.
Then DA = DE + AE = $x + 3$.

Substitute these into the ratio:

Cross-multiply:

Subtract $4x$ from both sides:

Divide by 5:

The length of DE is $\boxed{2.4 \text{ cm}}$.