Sure, let's break this down step-by-step:
## Step 1: Rewrite the terms that are subtracted as the addition of the opposite.
We are given the following polynomial expressions:
[tex]\[ -7g^4 + 4g^3 - 3g^2 + 5g - 3 \][/tex]
[tex]\[ -4g^4 - 3g^3 + 4g^2 + 5g + 3 \][/tex]
[tex]\[ g^2 + (-4g^4) + 5g + 9 + (-3g^3) + 3g^2 + (-6) \][/tex]
[tex]\[ -4g^4 + 4g^2 + 14g - 6 \][/tex]
Let’s rewrite any subtraction as addition of the opposite:
[tex]\[ -7g^4 + 4g^3 + (-3g^2) + 5g + (-3) \][/tex]
[tex]\[ -4g^4 + (-3g^3) + 4g^2 + 5g + 3 \][/tex]
[tex]\[ (-4g^4) + (-3g^3) + g^2 + 3g^2 + 5g + 9 + (-6) \][/tex]
[tex]\[ -4g^4 + 4g^2 + 14g + (-6) \][/tex]
## Step 2: Group like terms.
First, let's list out all terms grouped by their degree:
[tex]\[ \begin{align*}
-7g^4 &+ 4g^3 + (-3g^2) + 5g + (-3) \\
&+ (-4g^4) + (-3g^3) + 4g^2 + 5g + 3 \\
&+ (-4g^4) + (-3g^3) + g^2 + 3g^2 + 5g + 9 + (-6) \\
&+ (-4g^4) + 4g^2 + 14g + (-6)
\end{align} \][/tex]
Now, recombine them by sorting the terms according to their degree:
[tex]\[ \begin{align}
=& (-7g^4 + (-4g^4) + (-4g^4) + (-4g^4)) \\
&+ (4g^3 + (-3g^3) + (-3g^3) + (-3g^3)) \\
&+ ((-3g^2) + 4g^2 + g^2 + 3g^2 + 4g^2) \\
&+ \left( 5g + 5g + 5g + 14g \right) \\
&+ (-3 + 3 + 9 + (-6) + (-6))
\end{align*} \][/tex]
## Step 3: Combine like terms.
- For [tex]\( g^4 \)[/tex] terms:
[tex]\[ -7g^4 + (-4g^4) + (-4g^4) + (-4g^4) = -7g^4 - 12g^4 = -19g^4 \][/tex]
- For [tex]\( g^3 \)[/tex] terms:
[tex]\[ 4g^3 + (-3g^3) + (-3g^3) + (-3g^3) = 4g^3 - 9g^3 = -5g^3 \][/tex]
- For [tex]\( g^2 \)[/tex] terms:
[tex]\[ -3g^2 + 4g^2 + g^2 + 3g^2 + 4g^2 = 9g^2 - 3g^2 = 6g^2 \][/tex]
- For [tex]\( g \)[/tex] terms:
[tex]\[ 5g + 5g + 5g + 14g = 29g \][/tex]
- Constant terms:
[tex]\[ -3 + 3 + 9 - 6 - 6 = -3 \][/tex]
## Step 4: Write the resulting polynomial in standard form.
Combining all these results into a single polynomial, we get:
[tex]\[ -19g^4 - 5g^3 + 6g^2 + 29g - 3 \][/tex]
Thus, the sum of the given polynomial expressions is:
[tex]\[ -19g^4 - 5g^3 + 6g^2 + 29g - 3 \][/tex]
