To solve the equation \( \sqrt[3]{411x - 15} + 21 = 1 \), we first isolate the cube root term:
- Subtract 21 from both sides:
\[ \sqrt[3]{411x - 15} = 1 - 21 \] \[ \sqrt[3]{411x - 15} = -20 \]
- Cube both sides to eliminate the cube root:
\[ 411x - 15 = (-20)^3 \] \[ 411x - 15 = -8000 \]
- Add 15 to both sides:
\[ 411x = -8000 + 15 \] \[ 411x = -7985 \]
- Divide both sides by 411 to solve for \( x \):
\[ x = \frac{-7985}{411} \]
Now we can simplify \(-7985/411\). Since the numbers can be divided, we compute it:
- Divide \( 7985 \) by \( 411 \) (you can use a calculator for precision): \[ 7985 \div 411 \approx 19.4 \text{ (not an integer so we use it as is.)} \]
Thus, the exact value remains as:
\[ x = \frac{-7985}{411} \]
For simplification, check if \( 7985 \) and \( 411 \) share a common factor or divisibility. However, \( 7985 \) appears to be divisible by \( 5 \) and \( 11 \) but \( 411 \) is larger.
Without factorization finding brings \( 411 = 3 \times 137 \) and \( 7985 = 5 \times 79 \) without common factors.
Now, converting to an approximate fraction:
- Upon using a calculator for long division, we get to
So the answer is not included among the provided choices.
But if you want to see from provided, we have:
- Check \( x = \frac{-9}{11} \) or \( x = -10 \)
- They do not fit \( -7985/411 \)
In conclusion, we can remain with option \( \approx -19.4 \)
For the provided choices, none exactly matches our calculation.
Final \( x \) cannot be reflected from given response, which could match the \( -10 \) or non exactly resembling.
And thus direct return would require verification.
So provided responses do not yield exact math output but:
Final answer remains \( \frac{-7985}{411} \) which is \( ≈ -19.4 \). So please reconfirm inputs.