To solve the expression [tex]\(\left(2 a m + 5 a^2 b\right)^2\)[/tex], we need to expand it fully. Let's go through the expansion step by step.
1. Identify the binomial expression: The expression we need to expand is [tex]\(\left(2 a m + 5 a^2 b\right)^2\)[/tex].
2. Use the binomial square formula: The formula for the square of a binomial [tex]\((x + y)^2\)[/tex] is given by:
[tex]\[
(x + y)^2 = x^2 + 2xy + y^2
\][/tex]
Here, [tex]\(x = 2 a m\)[/tex] and [tex]\(y = 5 a^2 b\)[/tex].
3. Substitute the values into the binomial square formula:
Let's find each term in the formula:
- [tex]\(x^2 = (2 a m)^2\)[/tex]
- [tex]\(y^2 = (5 a^2 b)^2\)[/tex]
- [tex]\(2xy = 2 \cdot (2 a m) \cdot (5 a^2 b)\)[/tex]
4. Calculate each term:
- [tex]\(x^2 = (2 a m)^2 = 4 a^2 m^2\)[/tex]
- [tex]\(y^2 = (5 a^2 b)^2 = 25 a^4 b^2\)[/tex]
- [tex]\(2xy = 2 \cdot (2 a m) \cdot (5 a^2 b) = 2 \cdot 2 \cdot 5 \cdot a m \cdot a^2 b = 20 a^3 b m\)[/tex]
5. Combine the terms: Add all the calculated terms together:
[tex]\[
x^2 + 2xy + y^2 = 4 a^2 m^2 + 20 a^3 b m + 25 a^4 b^2
\][/tex]
6. Write the final expression: The expanded form of [tex]\(\left(2 a m + 5 a^2 b\right)^2\)[/tex] is:
[tex]\[
25 a^4 b^2 + 20 a^3 b m + 4 a^2 m^2
\][/tex]
So, the expanded expression is:
[tex]\[
\boxed{25 a^4 b^2 + 20 a^3 b m + 4 a^2 m^2}
\][/tex]
