Final answer:
After substituting each provided option into the expression 23x² + 173x + 12, none result in a negative outcome. Hence, none of the options presented (A. 6, B. 4 ¼, C. 1 ½, D. 2/3) support Richard's claim that the expression can be negative.
Step-by-step explanation:
The student is asking to determine if the expression 23x² + 173x + 12 can take on negative values for certain x values. To support Richard's claim, one would need to find a value of x where the entire expression results in a negative number. Using the provided options, we calculate the expression with each x value:
- For x=6, the expression is positive because all terms are positive when multiplied by a positive number like 6.
- For x=4.25, or 4 ¼, the expression is again positive for the same reason as stated above.
- For x=1.5, or 1 ½, even though the 173x term is substantial, the squares of 1.5 are positive and, added together with other positive terms, result in a positive value overall.
- For x=2/3, or approximately 0.67, the value of 23x² is less than 173x, but still not enough to make the entire 1 negative when added with 12.
After reviewing all possible options, none of the given x values result in the expression becoming negative; therefore, Richard's claim is not supported by any of the choices provided (A, B, C, D).