Math worksheet with polynomial factoring exercises and a ski-related joke.

Let’s solve this step by step.

We are given a polynomial to factor:
x⁴ + 8x³ - 20x²

Step 1: Look for the greatest common factor (GCF) in all terms.
All terms have at least x², so factor that out first:

→ x²(x² + 8x - 20)

Now we need to factor the quadratic inside: x² + 8x - 20

Step 2: Factor x² + 8x - 20
We need two numbers that:
- Multiply to -20
- Add to +8

Let’s list factor pairs of -20:
- 1 and -20 → sum = -19 ✘
- -1 and 20 → sum = 19 ✘
- 2 and -10 → sum = -8 ✘
- -2 and 10 → sum = 8 ✔ ← This works!

So, x² + 8x - 20 = (x - 2)(x + 10)

Wait — let’s check:
(x - 2)(x + 10) = x² + 10x - 2x - 20 = x² + 8x - 20 ✔️

But hold on — look back at our factored form:
We had x²(x² + 8x - 20) = x²(x - 2)(x + 10)

But now check the answer choices.

Look at choice S: x²(x + 10)(x - 2) ← That’s exactly what we got!

Also note: multiplication is commutative, so (x - 2)(x + 10) is same as (x + 10)(x - 2).

So the fully factored form is:
x²(x + 10)(x - 2)

That matches option S.

Now, according to the instructions:
“Each time the exercise number appears in the code, write this letter above it.”

The problem shown is labeled with a circled ⑤ — meaning this is Exercise #5.

In the “Answers for 1–7” column, find which letter corresponds to the correct factored form for #5.

Looking at the list:

⑤ → ?
We just found that the correct factored form is x²(x + 10)(x - 2), which is listed under S.

Check the left column:

Under “Answers for 1–7”, we see:

S → x²(x + 10)(x - 2) ← YES! That’s ours.

So for exercise #5, the letter is S.

Final Answer: S