We are given the function:

$f(x) = 2(x + 5)^2$

We need to expand and rewrite it in standard quadratic form:

$f(x) = 2x^2 + kx + 50$

Step 1: Expand $(x+5)^2$

$(x+5)^2 = x^2 + 10x + 25$

Step 2: Multiply by 2

$f(x) = 2(x^2 + 10x + 25)$

$= 2x^2 + 20x + 50$

Step 3: Compare with Standard Form

$2x^2 + 20x + 50$

Comparing coefficients, we see that $k = 20$.

Final Answer: $\boxed{20}$

Would you like a further breakdown of any step?

Related Questions:

1. How do you expand a squared binomial like $(x + 5)^2$?
2. What is the general formula for expanding $(a + b)^2$?
3. How do you determine coefficients in a quadratic function?
4. What is the vertex form of a quadratic equation?
5. How do you convert vertex form to standard form?

Tip: Always check your expanded expression by distributing properly to avoid common algebraic mistakes!