Certainly! Let's factor the given expression step-by-step:
Given the expression:
[tex]\[ q^4 + q^2r^2s + r^4s^2 \][/tex]
To factor this expression, we'll start by looking for common patterns that might simplify it. Notice that the expression consists of terms involving [tex]\( q \)[/tex] and [tex]\( r \)[/tex] raised to even powers, and the variable [tex]\( s \)[/tex] appearing in two of the terms. This suggests we might be able to use a specific factoring pattern involving squares. However, this expression doesn't fit a simple recognizable pattern like the sum or difference of squares, cubes, or other straightforward factorizations.
We can re-write the expression for clarity:
[tex]\[ q^4 + q^2r^2s + r^4s^2 \][/tex]
At this point, observe if there is a way to regroup or factor by common factors in any subset of terms. However, that doesn’t directly apply here. Therefore we must conclude that the given expression cannot be factored further using conventional algebraic methods.
Hence, our factorized form remains as:
[tex]\[ q^4 + q^2r^2s + r^4s^2 \][/tex]
This expression is already in its simplest form, indicating that no further factoring can be performed on it given the constraints of typical algebraic methods.
