Question 3

b = -2

To find the value of b, we need to expand the right-hand side of the equation \( (x + 3)(x - 5) \) and compare the coefficients with those in the quadratic expression \( x^2 + bx + c \).

A) -5

This value is incorrect because substituting b = -5 into the expression \( x^2 + bx + c \) would not match the linear coefficient obtained from expanding \( (x + 3)(x - 5) \), which is -2.

B) -2

This is the correct value for b. Expanding \( (x + 3)(x - 5) \) yields \( x^2 - 5x + 3x - 15 = x^2 - 2x - 15 \). Here, the coefficient of x is -2, which matches the form \( x^2 + bx + c \) where b = -2.

C) 3

This option is incorrect because if b were 3, the linear term in the expression \( x^2 + bx + c \) would be positive, which does not correspond with the -2 found in the expansion of \( (x + 3)(x - 5) \).

D) 8

This choice is invalid as well. If b were 8, the expression would produce a linear term of +8, which is inconsistent with the -2 from the expanded form of \( (x + 3)(x - 5) \).

E) 15

This option is also incorrect since a value of b equal to 15 would result in a significantly larger linear coefficient, which does not align with the -2 from the expansion of \( (x + 3)(x - 5) \).

Conclusion

The value of b is definitively -2, as it aligns perfectly with the linear coefficient obtained from the expansion of the given expression. All other options fail to satisfy this condition, making them incorrect in the context of the problem.