To solve the given expression [tex]\((x^2 c + 5 x^2 c^3)(2 x - 5 c^4)\)[/tex], we need to expand it step by step using the distributive property (also known as the FOIL method for binomials). This property tells us to multiply each term in the first parentheses by each term in the second parentheses:
Step 1: Write down the expression:
[tex]\[
(x^2 c + 5 x^2 c^3)(2 x - 5 c^4)
\][/tex]
Step 2: Distribute each term in the first parentheses to each term in the second parentheses.
Distribute [tex]\(x^2 c\)[/tex]:
[tex]\[
x^2 c \cdot 2 x + x^2 c \cdot (-5 c^4)
\][/tex]
Distribute [tex]\(5 x^2 c^3\)[/tex]:
[tex]\[
5 x^2 c^3 \cdot 2 x + 5 x^2 c^3 \cdot (-5 c^4)
\][/tex]
Step 3: Simplify each term resulting from the distribution.
[tex]\[
x^2 c \cdot 2 x = 2 x^3 c
\][/tex]
[tex]\[
x^2 c \cdot (-5 c^4) = -5 x^2 c^5
\][/tex]
[tex]\[
5 x^2 c^3 \cdot 2 x = 10 x^3 c^3
\][/tex]
[tex]\[
5 x^2 c^3 \cdot (-5 c^4) = -25 x^2 c^7
\][/tex]
Step 4: Combine all the terms together:
[tex]\[
2 x^3 c - 5 x^2 c^5 + 10 x^3 c^3 - 25 x^2 c^7
\][/tex]
The expanded expression is:
[tex]\[
-25 x^2 c^7 - 5 x^2 c^5 + 10 x^3 c^3 + 2 x^3 c
\][/tex]
Therefore, the expanded form of the given expression [tex]\((x^2 c + 5 x^2 c^3)(2 x - 5 c^4)\)[/tex] is:
[tex]\[
-25 c^7 x^2 - 5 c^5 x^2 + 10 c^3 x^3 + 2 c x^3
\][/tex]
