Factorize: (*x* + *y*)3 + 8*y*3

(*x* - 3*y*)(*x*2 + *y*2 + 4*y*2)

(*x*2 - *y*2 + 4*y*2)

(*x* + 3*y*) (*x*2 + 3*y*2)

(*x* + 3*y*) (*x*2 - *y*2)

Let's solve this step-by-step. The expression is:

\[
(x + y)^3 + 8y^3
\]

We recognize this as a **sum of cubes**. The general form of a sum of cubes is:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

Here, \(a = (x + y)\) and \(b = 2y\), since \(8y^3 = (2y)^3\). So, applying the formula:

\[
(x + y)^3 + (2y)^3 = \left((x + y) + 2y\right)\left((x + y)^2 - (x + y)(2y) + (2y)^2\right)
\]

Now simplify each part:

### First part:

\[
(x + y) + 2y = x + 3y
\]

### Second part:

\[
(x + y)^2 = x^2 + 2xy + y^2
\]

\[
(x + y)(2y) = 2xy + 2y^2
\]

\[
(2y)^2 = 4y^2
\]

Now substitute back into the expression:

\[
\left(x + 3y\right)\left(x^2 + 2xy + y^2 - (2xy + 2y^2) + 4y^2\right)
\]

Simplifying the terms inside the second bracket:

\[
x^2 + 2xy + y^2 - 2xy - 2y^2 + 4y^2 = x^2 + (y^2 - 2y^2 + 4y^2) = x^2 + 3y^2
\]

So, the factorized form is:

\[
(x + 3y)(x^2 + 3y^2)
\]

### Final Answer:
\[
(x + 3y)(x^2 + 3y^2)
\]

Would you like more details on any of the steps? Here are 5 related questions for practice:

1. How would you factorize \(x^3 + 27\)?
2. What is the factored form of \(a^3 - b^3\)?
3. Can you apply the sum of cubes formula to \(x^3 + 8\)?
4. How would you expand \((x + y)^3\)?
5. What is the general form for factoring a difference of cubes?

**Tip**: When factoring sums or differences of cubes, remember to recognize the cube terms and apply the correct formulas.

Learn how to factorize the polynomial expression (x + y)^3 + 8y^3 using the sum of cubes formula. This step-by-step guide demonstrates the key algebraic principles and provides practice questions for students in Grades 9-11.

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