Final answer:
The expression that we must add to 3x - 7 to result in x^2 + 4x - 1 is x^2 + x + 6. To answer the second question, we need to solve the quadratic inequality a^2 + 3a - 9 < 3 - 2a, which will provide the range for 'a' where 3 - 2a exceeds a^2 + a - 6.
Step-by-step explanation:
To answer the first student's question, we need to determine what must be added to 3x-7 to result in x^2+4x-1. We begin by setting up an equation:
3x - 7 + something = x^2 + 4x - 1.
To find out what that something is, we need to isolate it:
3x - 7 + something - 3x + 7 = x^2 + 4x - 1 - 3x + 7
Something = x^2 + x + 6
So the something we need to add is x^2 + x + 6.
To solve the second question, 'By how much does 3 - 2a exceed a^2 + a - 6?', we set up an inequality:
3 - 2a > a^2 + a - 6
Subtracting 3 - 2a from both sides gives us:
0 > a^2 + 3a - 9
This is a quadratic inequality, and its solution will give the range of values for which a^2 + 3a - 9 is less than 3 - 2a.