To add the expressions \( -3k + 4k^3 - 2 \) and \( 9k^2 - 3k^3 + 4k - 3 \), we will combine like terms.
Let's rewrite the expressions clearly:
Expression 1: \[ -3k + 4k^3 - 2 \]
Expression 2: \[ 9k^2 - 3k^3 + 4k - 3 \]
Now, combine the two expressions:
\[ (-3k + 4k^3 - 2) + (9k^2 - 3k^3 + 4k - 3) \]
Now we can group like terms:
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Cubic terms ( \(k^3\) ): \[ 4k^3 - 3k^3 = (4 - 3)k^3 = 1k^3 = k^3 \]
-
Quadratic terms ( \(k^2\) ): \[ 9k^2 = 9k^2 \]
-
Linear terms ( \(k\) ): \[ -3k + 4k = (-3 + 4)k = 1k = k \]
-
Constant terms: \[ -2 - 3 = -5 \]
Putting it all together, we get:
\[ k^3 + 9k^2 + k - 5 \]
So, the final result of adding the two expressions is:
\[ \boxed{k^3 + 9k^2 + k - 5} \]