Step-by-step solution for solving a word problem involving the area of a rectangle using polynomial factoring.

Let’s solve this step by step.

We are told:
- The length of a rectangle is 8 more than the width.
- The area is 240 square units.
- We let w = width.
- Then length = w + 8.
- Area = length × width → so:
240 = (w + 8) × w

That gives us the equation:
240 = w² + 8w

Now, to solve for w, we need to set the equation equal to zero (standard form for factoring):

Subtract 240 from both sides:
0 = w² + 8w - 240

Or written as:
w² + 8w - 240 = 0

Now we factor this quadratic.

We need two numbers that:
- Multiply to -240
- Add to +8

Let’s think of factor pairs of 240:

1 × 240
2 × 120
3 × 80
4 × 60
5 × 48
6 × 40
8 × 30 ← 30 - 8 = 22 → no
10 × 24 ← 24 - 10 = 14 → no
12 × 20 ← 20 - 12 = 8 → YES! But we need them to add to +8 and multiply to -240.

So if we use +20 and -12:
→ 20 × (-12) = -240
→ 20 + (-12) = +8 → Perfect!

So we can factor:

(w + 20)(w - 12) = 0

Set each factor equal to zero:

w + 20 = 0 → w = -20
w - 12 = 0 → w = 12

But width can’t be negative → so w = 12

Then length = w + 8 = 12 + 8 = 20

Check: Area = 12 × 20 = 240 → Correct!

Final Answer:
Width = 12, Length = 20