To solve the multiplication problem
[tex]\[
(3c^2 + 2d) \cdot (-5c^2 + d)
\][/tex]
we will use the distributive property of multiplication over addition, also known as the FOIL method (First, Outer, Inner, Last), to find each partial product.
#### Step-by-Step Solution:
1. First Terms:
[tex]\[
(3c^2) \cdot (-5c^2) = -15c^4
\][/tex]
2. Outer Terms:
[tex]\[
(3c^2) \cdot d = 3c^2d
\][/tex]
3. Inner Terms:
[tex]\[
2d \cdot (-5c^2) = -10c^2d
\][/tex]
4. Last Terms:
[tex]\[
2d \cdot d = 2d^2
\][/tex]
Now, we list all these partial products:
[tex]\[
-15c^4, \quad 3c^2d, \quad -10c^2d, \quad 2d^2
\][/tex]
Given the options, we identify which ones are present in our calculation:
1. \(2d^2\) - Yes, this was calculated as the product of the last terms.
2. \(3cd^3\) - No, there is no such term in the calculation.
3. \(-15c^4\) - Yes, this was calculated as the product of the first terms.
4. \(-10c^2d\) - Yes, this was calculated as the product of the inner terms.
5. \(-15c^2\) - No, there is no such term in the calculation.
6. \(3c^2d\) - Yes, this was calculated as the product of the outer terms.
Thus, the correct partial products from the given options are:
[tex]\[
-15c^4, \quad 3c^2d, \quad -10c^2d, \quad 2d^2
\][/tex]
