Compare Expressions: Which Options Equal 9a+5n in Algebraic Form

To solve this problem, we will simplify each expression given in options a through f, and compare each to the expression $9a + 5n$.

• a. $4(a+n)+5a\cdot n$
  Expanding, we have:
  $4a + 4n + 5an$.
  The presence of the term $5an$ makes it different from $9a + 5n$. Therefore, it is not equivalent.
• b. $3(a-n)+6a+8n$
  Expanding, we have:
  $3a - 3n + 6a + 8n$.
  Combining like terms, we get:
  $(3a + 6a) + (-3n + 8n) = 9a + 5n$.
  This matches $9a + 5n$.
• c. $20a-7n-11a+12n$
  Combining like terms, we have:
  $(20a - 11a) + (-7n + 12n) = 9a + 5n$.
  This also matches $9a + 5n$.
• d. $10a-n+6a-n$
  Combining like terms, we have:
  $(10a + 6a) + (-n - n) = 16a - 2n$.
  This is not equivalent to $9a + 5n$.
• e. $9(a+n)-\frac{4n}{2}$
  Expanding and simplifying, we have:
  $9a + 9n - 2n = 9a + 7n$.
  This is not $9a + 5n$.
• f. $3(a+n)(n-5)+24a-3n^2-3an+20n$
  Expanding $3(a+n)(n-5)$, we have:
  $3(an + n^2 - 5a - 5n) = 3an + 3n^2 - 15a - 15n$.
  The expression becomes:
  $(3an + 3n^2 - 15a - 15n) + 24a - 3n^2 - 3an + 20n$.
  Combining like terms, we find:
  $9a + 5n$.
  This matches $9a + 5n$.

Therefore, the expressions that are equal to $9a + 5n$ are options b, c, and f.

The correct answer choice is b, c, f.