Equivalent Expression Analysis: Comparing 45+5a+3ab+27b with Multiple Forms

To solve this problem, we will simplify each given expression and compare the results to the initial expression $45 + 5a + 3ab + 27b$.

Let's analyze each option:

• Option a: $(5+3b)(a+9)$
  Distribute: $= 5a + 45 + 3ba + 27b$
  Which rearranges to: $45 + 5a + 3ab + 27b$
  This matches the original expression.
• Option b: $45+\frac{2}{3}a+4\frac{1}{3}ab+(3a+27)b$
  Simplify: $= 45 + \frac{2}{3}a + \frac{13}{3}ab + 3ab + 27b$
  The terms do not simplify to match the original expression $45 + 5a + 3ab + 27b$.
• Option c: $2ab+5(9+a)+ab+\frac{9}{a}\cdot\frac{3ba}{1}$
  Simplify: Complete multiplication and distribution: $= 2ab + 45 + 5a + ab + 27b$
  Combine like terms: $= 45 + 5a + 3ab + 27b$
  This matches the original expression.
• Option d: $45+8ab+27b$
  This expression: $= 45 + 8ab + 27b$
  Depends on the situation with $b$, might match if conditions hold, but generally does not match unless specified.

After careful comparison, option a and c definitely represent the same value as the original expression, while option d depends on specific conditions regarding $b$.

The correct answer is: a, c, d (d. depends on b).