Alright, let's find the H.C.F. (Highest Common Factor) of the expressions [tex]\(4 P^3 q^2\)[/tex] and [tex]\(10 P^2 q^3\)[/tex].
### Step-by-Step Solution
1. Identify the Numerical Coefficients:
- The numerical coefficients of the given expressions are [tex]\(4\)[/tex] and [tex]\(10\)[/tex].
2. Find the GCD of the Numerical Coefficients:
- The greatest common divisor (GCD) of [tex]\(4\)[/tex] and [tex]\(10\)[/tex] is [tex]\(2\)[/tex].
3. Examine the Variables and Their Powers:
- For the variable [tex]\(P\)[/tex]:
- In the first expression, [tex]\(P\)[/tex] is raised to the power [tex]\(3\)[/tex] (i.e., [tex]\(P^3\)[/tex]).
- In the second expression, [tex]\(P\)[/tex] is raised to the power [tex]\(2\)[/tex] (i.e., [tex]\(P^2\)[/tex]).
- The minimum power of [tex]\(P\)[/tex] in these expressions is [tex]\(2\)[/tex].
- For the variable [tex]\(q\)[/tex]:
- In the first expression, [tex]\(q\)[/tex] is raised to the power [tex]\(2\)[/tex] (i.e., [tex]\(q^2\)[/tex]).
- In the second expression, [tex]\(q\)[/tex] is raised to the power [tex]\(3\)[/tex] (i.e., [tex]\(q^3\)[/tex]).
- The minimum power of [tex]\(q\)[/tex] in these expressions is [tex]\(2\)[/tex].
4. Combine the GCD of the Numerical Coefficients with the Minimum Powers of the Variables:
- The H.C.F. will be the product of the GCD of the numerical coefficients and the variables raised to their minimum powers.
- Hence, the H.C.F. is [tex]\(2 \cdot P^2 \cdot q^2\)[/tex].
Thus, the H.C.F. of the expressions [tex]\(4 P^3 q^2\)[/tex] and [tex]\(10 P^2 q^3\)[/tex] is [tex]\(2 P^2 q^2\)[/tex].
### Conclusion
Among the given choices, the correct one is:
[tex]\[
\text{d) } 2 P^2 q^2
\][/tex]
