Let's solve each part step by step.

(i) $(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$

This expression is a known identity for the expansion of $(a + b + c)$.

It simplifies to:

$(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = (a + b + c)^3 - 3abc$

This is a standard formula derived from the expansion of the cubic expression $(a + b + c)^3$.

Thus, the simplified expression is:

$(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = (a + b + c)^3 - 3abc$

---

(ii) $(5x^3 - 7x^2y - xy^2 + 3y^3)(5x + 3y)$

We'll expand this expression using the distributive property (also called the FOIL method in binomials):

$(5x^3 - 7x^2y - xy^2 + 3y^3)(5x + 3y)$

Now distribute each term from the first bracket to both terms in the second bracket:

• $5x^3 \cdot 5x = 25x^4$
• $5x^3 \cdot 3y = 15x^3y$
• $-7x^2y \cdot 5x = -35x^3y$
• $-7x^2y \cdot 3y = -21x^2y^2$
• $-xy^2 \cdot 5x = -5x^2y^2$
• $-xy^2 \cdot 3y = -3xy^3$
• $3y^3 \cdot 5x = 15xy^3$
• $3y^3 \cdot 3y = 9y^4$

Now, combine like terms:

$25x^4 + (15x^3y - 35x^3y) + (-21x^2y^2 - 5x^2y^2) + (-3xy^3 + 15xy^3) + 9y^4$

Simplifying further:

$25x^4 - 20x^3y - 26x^2y^2 + 12xy^3 + 9y^4$

Thus, the final expanded expression is:

$25x^4 - 20x^3y - 26x^2y^2 + 12xy^3 + 9y^4$

---

Do you want further details on any step?

5 Related Questions:

1. How do you derive the identity for $(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$?
2. Can the expression $(5x^3 - 7x^2y - xy^2 + 3y^3)(5x + 3y)$ be factored in any other way?
3. What are some typical identities involving cubic expressions?
4. How is the FOIL method applied to larger polynomials?
5. What are common mistakes when expanding polynomial expressions?

Tip:

When simplifying expressions, look for patterns like common identities or symmetry to reduce calculation complexity.