To find the total volume of the two stacked cubes, we need to first calculate the volume of each cube individually.
1. Volume of the first cube:
- The side length of the first cube is [tex]\(4p\)[/tex].
- The volume [tex]\(V_1\)[/tex] of a cube is given by the formula [tex]\(s^3\)[/tex], where [tex]\(s\)[/tex] is the side length.
- Therefore, the volume of the first cube is:
[tex]\[
V_1 = (4p)^3
\][/tex]
- Calculating this, we get:
[tex]\[
V_1 = 64p^3
\][/tex]
2. Volume of the second cube:
- The side length of the second cube is [tex]\(2q^2\)[/tex].
- The volume [tex]\(V_2\)[/tex] of a cube is given by the formula [tex]\(s^3\)[/tex], where [tex]\(s\)[/tex] is the side length.
- Therefore, the volume of the second cube is:
[tex]\[
V_2 = (2q^2)^3
\][/tex]
- Calculating this, we get:
[tex]\[
V_2 = 8q^6
\][/tex]
3. Total volume:
- The total volume of both cubes stacked is the sum of the individual volumes:
[tex]\[
V_{\text{total}} = V_1 + V_2
\][/tex]
- Therefore, the total volume is:
[tex]\[
V_{\text{total}} = 64p^3 + 8q^6
\][/tex]
We are asked to express the total volume in factored form. Given the factored forms in the options, let's see which one matches [tex]\(64p^3 + 8q^6\)[/tex]:
[tex]\[
64p^3 + 8q^6 = 8(8p^3 + q^6)
\][/tex]
None of the provided options factor the polynomial [tex]\(64p^3 + 8q^6\)[/tex] in a straightforward manner. However, examining the forms, we can compare coefficients:
- We take the first expression and expand it to compare:
[tex]\[
(4p + 2q^2)(16p^2 - 8pq^2 + 4q^4)
\][/tex]
Expanding this:
[tex]\[
(4p + 2q^2)(16p^2 - 8pq^2 + 4q^4) = 4p(16p^2 - 8pq^2 + 4q^4) + 2q^2(16p^2 - 8pq^2 + 4q^4)
\][/tex]
[tex]\[
= 64p^3 - 32p^2q^2 + 16pq^4 + 32p^2q^2 - 16pq^4 + 8q^6
\][/tex]
[tex]\[
= 64p^3 + 8q^6
\][/tex]
This is equal to our total volume [tex]\(64p^3 + 8q^6\)[/tex].
Thus, the factored form that correctly represents the total volume is:
[tex]\[
\left(4 p+2 q^2\right)\left(16 p^2-8 p q^2+4 q^4\right)
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{\left(4 p+2 q^2\right)\left(16 p^2-8 p q^2+4 q^4\right)}
\][/tex]
