To solve the expression [tex]\(\left(2 p q^3 - 5 p^4 q^2\right)^2\)[/tex], let's expand and simplify it step-by-step.
### Step 1: Distribute the Squared Term
We need to expand the square of the binomial expression:
[tex]\[
\left(2 p q^3 - 5 p^4 q^2\right)^2
\][/tex]
### Step 2: Use the Binomial Theorem Formula
The square of a binomial [tex]\((a - b)^2\)[/tex] can be expanded using the formula:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
Here, [tex]\(a = 2 p q^3\)[/tex] and [tex]\(b = 5 p^4 q^2\)[/tex].
### Step 3: Calculate Each Term Separately
Now we will expand the terms individually:
#### First Term: [tex]\((2 p q^3)^2\)[/tex]
[tex]\[
(2 p q^3)^2 = 4 p^2 q^6
\][/tex]
#### Second Term: [tex]\(-2 \cdot (2 p q^3) \cdot (5 p^4 q^2)\)[/tex]
[tex]\[
-2 (2 p q^3) (5 p^4 q^2) = -20 p^5 q^5
\][/tex]
#### Third Term: [tex]\((5 p^4 q^2)^2\)[/tex]
[tex]\[
(5 p^4 q^2)^2 = 25 p^8 q^4
\][/tex]
### Step 4: Combine the Terms
Finally, we combine all the terms obtained in the previous step:
[tex]\[
\left(2 p q^3 - 5 p^4 q^2\right)^2 = 4 p^2 q^6 - 20 p^5 q^5 + 25 p^8 q^4
\][/tex]
### Step 5: Arranging the Terms
We can arrange the terms in descending order of the powers of [tex]\(p\)[/tex] and [tex]\(q\)[/tex]:
[tex]\[
25 p^8 q^4 - 20 p^5 q^5 + 4 p^2 q^6
\][/tex]
Hence, the expanded form of the expression [tex]\(\left(2 p q^3 - 5 p^4 q^2\right)^2\)[/tex] is:
[tex]\[
25 p^8 q^4 - 20 p^5 q^5 + 4 p^2 q^6
\][/tex]
