Expand (a+4b)(2c+a): Binomial Multiplication with Three Variables

To solve this problem, we'll follow these steps:

• Step 1: Distribute each term in the first binomial $(a + 4b)$ to each term in the second binomial $(2c + a)$.
• Step 2: Simplify the expanded terms by combining like terms (if any).

Now, let's work through each step:
Step 1: Distribute $a$ from the first binomial to each term in the second binomial: - Distribute $a$ to $2c$: $a \cdot 2c = 2ac$ - Distribute $a$ to $a$: $a \cdot a = a^2$
Step 2: Distribute $4b$ from the first binomial to each term in the second binomial: - Distribute $4b$ to $2c$: $4b \cdot 2c = 8bc$ - Distribute $4b$ to $a$: $4b \cdot a = 4ab$
Now, combining all these results gives us:
$2ac + a^2 + 8bc + 4ab$

Therefore, the expanded form of the expression $(a+4b)(2c+a)$ is $2ac + a^2 + 8bc + 4ab$.