To simplify the expression \( 4x^2(5x^3+4x+1)−9x^5+3x^2+4 \), we will first distribute \( 4x^2 \) through the polynomial inside the parentheses.
Starting with:
\[ 4x^2(5x^3 + 4x + 1) \]
Distributing \( 4x^2 \):
\[ = 4x^2 \cdot 5x^3 + 4x^2 \cdot 4x + 4x^2 \cdot 1 \]
This simplifies to:
\[ = 20x^5 + 16x^3 + 4x^2 \]
Next, we substitute this into the original expression:
\[ 20x^5 + 16x^3 + 4x^2 - 9x^5 + 3x^2 + 4 \]
Now, we can combine like terms. The \( x^5 \) terms are \( 20x^5 \) and \( -9x^5 \):
\[ 20x^5 - 9x^5 = 11x^5 \]
Next, combining the \( x^3 \) terms, we only have \( 16x^3 \):
\[ = 16x^3 \]
Then, we combine the \( x^2 \) terms, which are \( 4x^2 \) and \( 3x^2 \):
\[ 4x^2 + 3x^2 = 7x^2 \]
Lastly, we have the constant term, which is \( 4 \).
Putting it all together, the simplified expression is:
\[ 11x^5 + 16x^3 + 7x^2 + 4 \]
In this expression, the first term is \( 11x^5 \) and the last term is \( 4 \). Thus, the first and last terms of the simplified expression are:
First term: \( 11x^5 \)
Last term: \( 4 \)