One of the factors of 2x3 + 42x 2 + 208x is x + b, where b is a positive constant. What is the smallest possible value of b ?

Explanation

The given expression is 2x3 + 42x2 + 208x. First, factor out the common factor of 2x: this gives 2x(x2 + 21x + 104). Next, factor the quadratic x2 + 21x + 104. To do this, find two numbers that add up to 21 and multiply to 104. These numbers are 8 and 13. Therefore, x2 + 21x + 104 can be factored as (x + 8)(x + 13). Thus, the original expression factors as 2x(x + 8)(x + 13). The factors of the form x + b are x + 8 and x + 13, with the smallest b being 8. Therefore, the correct answer is 8. Options 2, 4, and 6 do not align with the factors obtained from the correct factoring process.