To simplify the expression [tex]\((x + 2y)(x^2 - 3y)(4x + y)\)[/tex], we will follow a step-by-step process.
### Step 1: Expand the first two factors
First, let's expand the product [tex]\((x + 2y)(x^2 - 3y)\)[/tex].
[tex]\[
(x + 2y)(x^2 - 3y) = x(x^2 - 3y) + 2y(x^2 - 3y)
\][/tex]
Distribute [tex]\(x\)[/tex] and [tex]\(2y\)[/tex] over [tex]\(x^2 - 3y\)[/tex]:
[tex]\[
x(x^2 - 3y) + 2y(x^2 - 3y) = x^3 - 3xy + 2yx^2 - 6y^2
\][/tex]
Rearrange and combine like terms:
[tex]\[
x^3 + 2yx^2 - 3xy - 6y^2
\][/tex]
This simplifies to:
[tex]\[
x^3 + 2x^2y - 3xy - 6y^2
\][/tex]
### Step 2: Expand the result with the third factor
Next, multiply this result by the third factor [tex]\((4x + y)\)[/tex].
[tex]\[
(x^3 + 2x^2y - 3xy - 6y^2)(4x + y)
\][/tex]
Distribute each term of the first polynomial over the second polynomial:
[tex]\[
x^3(4x + y) + 2x^2y(4x + y) - 3xy(4x + y) - 6y^2(4x + y)
\][/tex]
Now, we expand each product separately:
[tex]\[
x^3(4x + y) = 4x^4 + x^3y
\][/tex]
[tex]\[
2x^2y(4x + y) = 8x^3y + 2x^2y^2
\][/tex]
[tex]\[
3xy(4x + y) = 12x^2y + 3xy^2
\][/tex]
[tex]\[
-6y^2(4x + y) = -24xy^2 - 6y^3
\][/tex]
### Step 3: Combine all terms together
Add all these expanded terms together:
[tex]\[
4x^4 + x^3y + 8x^3y + 2x^2y^2 - 12x^2y - 3xy^2 - 24xy^2 - 6y^3
\][/tex]
### Step 4: Simplify by combining like terms
Combine the like terms:
[tex]\[
4x^4 + (x^3y + 8x^3y) + 2x^2y^2 - 12x^2y - (3xy^2 + 24xy^2) - 6y^3
\][/tex]
Simplify the coefficients:
[tex]\[
4x^4 + 9x^3y + 2x^2y^2 - 12x^2y - 27xy^2 - 6y^3
\][/tex]
So, the simplified expression is:
[tex]\[
4x^4 + 9x^3y + 2x^2y^2 - 12x^2y - 27xy^2 - 6y^3
\][/tex]
