To add the expressions \( -3k + 4k^3 - 2 \) and \( 9k^2 - 3k^3 + 4k - 3 \), we will combine like terms.
Step 1: Write the expressions
Expression 1:
\[
-3k + 4k^3 - 2
\]
Expression 2:
\[
9k^2 - 3k^3 + 4k - 3
\]
Step 2: Combine like terms
Now we group and combine the terms based on their degree (i.e., \( k^3 \), \( k^2 \), and \( k \)):
-
\( k^3 \) terms: \[ 4k^3 - 3k^3 = (4 - 3)k^3 = 1k^3 \]
-
\( k^2 \) terms: \[ 0 + 9k^2 = 9k^2 \]
-
\( k \) terms: \[ -3k + 4k = (-3 + 4)k = 1k \]
-
Constant terms: \[ -2 - 3 = -5 \]
Step 3: Write the final result
Combining all of these results, we have:
\[ 1k^3 + 9k^2 + 1k - 5 \]
Thus, we can fill in the blanks as follows:
\[ 1k^3 + 9k^2 + 1k - 5 \]
So the final answer is:
\[ \boxed{1k^3 + 9k^2 + 1k - 5} \]