To form the quadratic equation from the given statement, let's break it down step by step.
The statement is: "the product of negative six and a number squared, plus the product of five and the same number is thirty-one."
Let the variable \( x \) represent "the number."
- Product of negative six and a number squared: This translates to \( -6x^2 \).
- Product of five and the same number: This translates to \( 5x \).
- The equation states this total is equal to thirty-one: So we have: \[ -6x^2 + 5x = 31 \]
Now, rearranging this equation to standard form: \[ -6x^2 + 5x - 31 = 0 \]
However, this is not one of the options provided. But the equivalent expression in terms of equality to \( 31 \) would be \( -6x^2 + 5x - 31 = 0 \).
The closest form among the given choices that reflects this equation accurately is:
-6x² + 5x = 31.
Hence, the equivalent quadratic equation is: -6x² + 5x = 31.