Final answer:
The polynomial with roots at -1 and 5 is 6x² - 7x - 5. This is found by examining which multiple choice option expands from the factors (x + 1) and (x - 5), which represent the given roots. Option a is the correct answer as it corresponds to a multiple of the expanded form x² - 4x - 5.
Step-by-step explanation:
The student is asking which polynomial has roots at -1 and 5. To solve this, we look for a polynomial that can be factored or is already in a factored form that includes (x + 1)(x - 5). The roots of a polynomial are the values of x that make the polynomial equal to zero. Since the question provides us with multiple choice answers, we only need to check which option results in those roots.
A polynomial with roots at -1 and 5 would have factors corresponding to these roots i.e., (x + 1) for the root at -1 and (x - 5) for the root at 5. If we expand these two factors, we get the polynomial: x² - 4x - 5. Now, we need to look at which option corresponds to a multiple of this polynomial.
Looking at our options, we can eliminate those that don't fit the expanded form of our factors. Option a, 6x² - 7x - 5, is the only choice that seems to be a multiple of x² - 4x - 5, since multiplying by 6 would not alter the roots, only the coefficient of the quadratic term. Thus, the correct answer is option a 6x² - 7x - 5