Final answer:
To factor the algebraic expression '182r', we identify the factors of 182 as 13 and 14. Evaluating the given options, we conclude that the correct factorization is option b) (13r+9)(14r+9), as the signs must be positive to match the middle term of the expression.
Step-by-step explanation:
The goal here is to factor the given algebraic expression, which seems to have a typographical error in part of it. We will focus on the recognizable part of the expression '182r'. This coefficient suggests that each factor of the complete expression might include a term that multiplies to 182. Now, 182 can be factored into 14 and 13, which gives us a starting point. Since we have several options with combinations of 13r and 14r with either +9 or -9, we need to identify the correct signs that multiply to give the final terms of the full expression.
By examining the options provided, we can use the fact that the signs of the middle term in a quadratic expression (when written as '(ax + b)(cx + d)' would add to 'ac' and 'bd' to get the middle term. In this case, we have '+9' from both factors. Therefore, the factors of the expression should include '+9' in both parentheses since there is no negative part mentioned in the recognizable part of the expression. This leads us to conclude option b) (13r+9)(14r+9) is the correct factorization.