To simplify the expression \(4x^2(5x^3 + 4x + 1) - 9x^5 + 3x^2 + 4\), we will first distribute \(4x^2\) across the terms inside the parentheses.
Starting with the distribution:
\[ 4x^2 \cdot 5x^3 = 20x^5, \] \[ 4x^2 \cdot 4x = 16x^3, \] \[ 4x^2 \cdot 1 = 4x^2. \]
Now we can rewrite the expression as:
\[ 20x^5 + 16x^3 + 4x^2 - 9x^5 + 3x^2 + 4. \]
Next, we will combine the like terms. First, we combine the \(x^5\) terms:
\[ 20x^5 - 9x^5 = 11x^5. \]
Now we combine the \(x^3\) terms:
\[ 16x^3. \]
Next, we combine the \(x^2\) terms:
\[ 4x^2 + 3x^2 = 7x^2. \]
Finally, the constant term remains:
\[ 4. \]
Putting it all together, we have:
\[ 11x^5 + 16x^3 + 7x^2 + 4. \]
Now we can identify the first and last terms of the simplified expression:
- First term: \(11x^5\)
- Last term: \(4\)
Thus, the first term is \(11x^5\) and the last term is \(4\).