To expand $(x + 5)^3$, you can use the binomial theorem or manually expand it by multiplying the binomial three times. Here's the step-by-step process using the binomial theorem:

Binomial Theorem

The binomial theorem states: $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ For $(x + 5)^3$, $a = x$, $b = 5$, and $n = 3$. Plugging these into the formula:

$(x + 5)^3 = \binom{3}{0}x^3(5)^0 + \binom{3}{1}x^2(5)^1 + \binom{3}{2}x^1(5)^2 + \binom{3}{3}x^0(5)^3$

Now, calculate each term:

1. $\binom{3}{0}x^3(5)^0 = 1 \cdot x^3 \cdot 1 = x^3$
2. $\binom{3}{1}x^2(5)^1 = 3 \cdot x^2 \cdot 5 = 15x^2$
3. $\binom{3}{2}x^1(5)^2 = 3 \cdot x \cdot 25 = 75x$
4. $\binom{3}{3}x^0(5)^3 = 1 \cdot 1 \cdot 125 = 125$

Combine the terms:

$x^3 + 15x^2 + 75x + 125$

So, the expanded form of $(x + 5)^3$ is: $x^3 + 15x^2 + 75x + 125$

Would you like more details on this process or have any questions?

Here are five related questions:

1. How do you expand $(x - 3)^4$ using the binomial theorem?
2. What is the general form of the binomial theorem for any power $n$?
3. How can you factor a cubic expression like $x^3 + 15x^2 + 75x + 125$?
4. How do you apply the binomial theorem to expressions with more complex terms, such as $(2x - 7)^3$?
5. What are the real-world applications of the binomial theorem?

Tip: When using the binomial theorem, make sure you calculate each binomial coefficient correctly to avoid errors in your expansion.