Sure! Let's solve the given expression [tex]\(\left(-6x - 7y^2\right)^2\)[/tex] step-by-step.
Step 1: Understand the given expression.
We are given [tex]\((-6x - 7y^2)^2\)[/tex] and need to expand this expression.
Step 2: Apply the binomial theorem.
The binomial theorem states that for any two terms [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[(a + b)^2 = a^2 + 2ab + b^2\][/tex]
In our case, [tex]\(a = -6x\)[/tex] and [tex]\(b = -7y^2\)[/tex]. So, we can use this theorem to expand [tex]\((-6x - 7y^2)^2\)[/tex].
Step 3: Compute the individual terms:
1. [tex]\( a^2 = (-6x)^2 \)[/tex]
2. [tex]\( 2ab = 2 \cdot (-6x) \cdot (-7y^2) \)[/tex]
3. [tex]\( b^2 = (-7y^2)^2 \)[/tex]
Let's compute these separately:
1. Computing [tex]\( a^2 \)[/tex]:
[tex]\[
(-6x)^2 = (-6)^2 \cdot x^2 = 36x^2
\][/tex]
2. Computing [tex]\( 2ab \)[/tex]:
[tex]\[
2 \cdot (-6x) \cdot (-7y^2) = 2 \cdot 6 \cdot 7 \cdot x \cdot y^2 = 84xy^2
\][/tex]
3. Computing [tex]\( b^2 \)[/tex]:
[tex]\[
(-7y^2)^2 = (-7)^2 \cdot (y^2)^2 = 49y^4
\][/tex]
Step 4: Combine the results:
Now, put all these computed terms together:
[tex]\[
(-6x - 7y^2)^2 = 36x^2 + 84xy^2 + 49y^4
\][/tex]
So, the expanded form of [tex]\(\left(-6x - 7y^2\right)^2\)[/tex] is:
[tex]\[
36x^2 + 84xy^2 + 49y^4
\][/tex]
