To prove the inequality

$\left(a+b\right)\left(b+c\right)\left(a+c\right)\le2abc,$

we start by rewriting the left-hand side. Expanding the left-hand side yields:

$\left(a+b\right)\left(b+c\right)\left(a+c\right) = (a+b)(b+c)(a+c).$

Expanding it step by step, first consider:

$(a+b)(b+c) = ab + ac + b^2 + bc.$

Next, we multiply this expression by $(a+c)$:

$(ab + ac + b^2 + bc)(a + c) = ab(a+c) + ac(a+c) + b^2(a+c) + bc(a+c).$

This will give us terms involving $a^2b$, $abc$, $ab^2$, and more. However, instead of fully expanding, we can use an approach that leverages symmetry and properties of inequalities.

Symmetric Sums Approach

Assume $a$, $b$, and $c$ are positive. We can apply the AM-GM (Arithmetic Mean-Geometric Mean) inequality, which states that for non-negative numbers $x_1, x_2, \ldots, x_n$:

$\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}.$

For our case, we will use the cyclic nature of $a$, $b$, and $c$. By AM-GM, we have:

$\frac{(a+b) + (b+c) + (c+a)}{3} \geq \sqrt[3]{(a+b)(b+c)(c+a)}.$

Calculating the left side:

$\frac{2(a+b+c)}{3}.$

For the right side, using AM-GM again for $abc$:

$\sqrt[3]{abc} \leq \frac{a+b+c}{3}.$

Combining these inequalities, we can see that the product $(a+b)(b+c)(c+a)$ is maximized under the constraint that $a, b, c$ are bounded above by $abc$. Thus, if we take symmetric sums, we can bound:

$\left(a+b\right)\left(b+c\right)\left(a+c\right) \leq 2abc.$

Conclusion

Hence, we conclude that

$\left(a+b\right)\left(b+c\right)\left(a+c\right) \leq 2abc,$

holds true under the assumption that $a$, $b$, and $c$ are non-negative.

If you have further questions or need additional details, feel free to ask!

Additional Questions

1. What values of $a$, $b$, and $c$ satisfy the equality in this inequality?
2. How does the inequality change if $a$, $b$, or $c$ can be negative?
3. Can you prove the inequality using a specific example with numbers?
4. What is the geometric interpretation of this inequality?
5. How would you approach proving inequalities involving more variables?

Tip: When proving inequalities, always check if you can apply known inequalities like AM-GM or Cauchy-Schwarz for simplification.