To solve the problem, we need to find the other polynomial when given the sum of two polynomials and one of the polynomials. Let's break this down into clear, step-by-step calculations.
### Step 1: Represent the given information:
- The sum of two polynomials: [tex]\( -y z^2 - 3 z^2 - 4 y + 4 \)[/tex]
- One of the polynomials: [tex]\( y - 4 y z^2 - 3 \)[/tex]
### Step 2: Represent the unknown polynomial with a variable:
Let the unknown polynomial be [tex]\( P_{\text{unknown}}(y, z) \)[/tex].
### Step 3: Set up the equation:
According to the problem, the sum of the given polynomial and the unknown polynomial equals the provided polynomial sum:
[tex]\[ (P_{\text{unknown}}(y, z)) + (y - 4 y z^2 - 3) = -y z^2 - 3 z^2 - 4 y + 4 \][/tex]
### Step 4: Isolate the unknown polynomial:
To find the unknown polynomial, we need to isolate it on one side of the equation:
[tex]\[ P_{\text{unknown}}(y, z) = -y z^2 - 3 z^2 - 4 y + 4 - (y - 4 y z^2 - 3) \][/tex]
### Step 5: Distribute and simplify:
Now, simplify the right-hand side:
[tex]\[ P_{\text{unknown}}(y, z) = -y z^2 - 3 z^2 - 4 y + 4 - y + 4 y z^2 + 3 \][/tex]
Combine like terms:
- Combine [tex]\( -y z^2 \)[/tex] and [tex]\( 4 y z^2 \)[/tex]:
[tex]\[ -y z^2 + 4 y z^2 = 3 y z^2 \][/tex]
- Combine [tex]\( -3 z^2 \)[/tex] with no other [tex]\( z^2 \)[/tex] terms:
[tex]\[ -3 z^2 \][/tex]
- Combine [tex]\( -4 y \)[/tex] and [tex]\( -y \)[/tex]:
[tex]\[ -4 y - y = -5 y \][/tex]
- Combine [tex]\( 4 \)[/tex] and [tex]\( 3 \)[/tex]:
[tex]\[ 4 + 3 = 7 \][/tex]
Putting it all together:
[tex]\[ P_{\text{unknown}}(y, z) = 3 y z^2 - 3 z^2 - 5 y + 7 \][/tex]
### Conclusion:
The other polynomial is:
[tex]\[
\boxed{3 y z^2 - 3 z^2 - 5 y + 7}
\][/tex]
This matches one of the given options.
