Expanding and Comparing Polynomials: (x+2)(x3)(x+5) and x^3 + ax^2  11x + b
This article explores the relationship between the expanded form of the polynomial (x+2)(x3)(x+5) and the polynomial x^3 + ax^2  11x + b. Our goal is to determine the values of a and b that make the two expressions identical.
Expanding the Polynomial
First, we expand the product (x+2)(x3)(x+5) using the distributive property (or FOIL method):

Expand (x+2)(x3): (x+2)(x3) = x²  x  6

Multiply the result by (x+5): (x²  x  6)(x+5) = x³ + 4x²  11x  30
Therefore, the expanded form of (x+2)(x3)(x+5) is x³ + 4x²  11x  30.
Comparing Coefficients
Now we compare the expanded form with x³ + ax²  11x + b:
 x³ coefficient: Both expressions have a coefficient of 1 for the x³ term.
 x² coefficient: The expanded form has a coefficient of 4 for the x² term, while the other expression has a coefficient of a. Therefore, a = 4.
 x coefficient: Both expressions have a coefficient of 11 for the x term.
 Constant term: The expanded form has a constant term of 30, while the other expression has a constant term of b. Therefore, b = 30.
Conclusion
We conclude that the polynomial (x+2)(x3)(x+5) is identical to x³ + 4x²  11x  30. Therefore, a = 4 and b = 30.