Sure, let's work through the problem step by step.
The given expression is:
[tex]$(x-4)(2x + 3y)^2$[/tex]
To expand this expression, we need to use the distributive property and the binomial expansion. Here’s the detailed step-by-step solution:
1. First Step: Identify the binomial inside the square:
[tex]$(2x + 3y)^2$[/tex]
2. Second Step: Expand the binomial using the formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex]:
[tex]$(2x + 3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2$[/tex]
Calculating each term:
[tex]\[
(2x)^2 = 4x^2
\][/tex]
[tex]\[
2(2x)(3y) = 12xy
\][/tex]
[tex]\[
(3y)^2 = 9y^2
\][/tex]
So, the expanded form of [tex]\((2x + 3y)^2\)[/tex] is:
[tex]\[
4x^2 + 12xy + 9y^2
\][/tex]
3. Third Step: Multiply this result by [tex]\((x - 4)\)[/tex]:
[tex]$ (x - 4)(4x^2 + 12xy + 9y^2) $[/tex]
4. Fourth Step: Distribute [tex]\((x - 4)\)[/tex] to each term inside the parentheses:
[tex]\[
x \cdot (4x^2 + 12xy + 9y^2) - 4 \cdot (4x^2 + 12xy + 9y^2)
\][/tex]
Expanding these terms:
[tex]\[
x \cdot 4x^2 = 4x^3
\][/tex]
[tex]\[
x \cdot 12xy = 12x^2y
\][/tex]
[tex]\[
x \cdot 9y^2 = 9xy^2
\][/tex]
And,
[tex]\[
-4 \cdot 4x^2 = -16x^2
\][/tex]
[tex]\[
-4 \cdot 12xy = -48xy
\][/tex]
[tex]\[
-4 \cdot 9y^2 = -36y^2
\][/tex]
5. Combine all the terms:
[tex]\[
4x^3 + 12x^2y + 9xy^2 - 16x^2 - 48xy - 36y^2
\][/tex]
So the expanded form of [tex]\((x - 4)(2x + 3y)^2\)[/tex] is:
[tex]\[
4x^3 + 12x^2y - 16x^2 + 9xy^2 - 48xy - 36y^2
\][/tex]
And that’s your final expanded expression!
