Sure, let's go through the process of adding the given polynomials step by step:
We are given two polynomials:
[tex]\[ P_1(t) = -t^4 - 5t^3 - 10t^2 \][/tex]
[tex]\[ P_2(t) = 9t^3 + 3t^2 - 1 \][/tex]
We need to add these polynomials. Let's align the terms with respect to the powers of [tex]\( t \)[/tex]:
[tex]\[
\begin{array}{rcccc}
P_1(t) & = & -t^4 & - 5t^3 & - 10t^2 & + 0t & + 0 \\
P_2(t) & = & 0t^4 & + 9t^3 & + 3t^2 & + 0t & - 1 \\
\end{array}
\][/tex]
Now, add the corresponding terms together:
1. The [tex]\( t^4 \)[/tex] term:
[tex]\[ -t^4 + 0t^4 = -t^4 \][/tex]
2. The [tex]\( t^3 \)[/tex] term:
[tex]\[ -5t^3 + 9t^3 = 4t^3 \][/tex]
3. The [tex]\( t^2 \)[/tex] term:
[tex]\[ -10t^2 + 3t^2 = -7t^2 \][/tex]
4. The [tex]\( t \)[/tex] term:
[tex]\[ 0t + 0t = 0t \][/tex]
5. The constant term:
[tex]\[ 0 - 1 = -1 \][/tex]
Combining these results, the sum of the polynomials in standard form is:
[tex]\[ -t^4 + 4t^3 - 7t^2 - 1 \][/tex]
So, the expanded polynomial in standard form is:
[tex]\[
\boxed{-t^4 + 4t^3 - 7t^2 - 1}
\][/tex]
