Expand and simplify (2x + 5)^2

Expand and simplify (2x + 5)^2

The Correct Answer and Explanation is :

The correct answer is: {4x^2 + 20x + 25}

To expand and simplify ((2x + 5)^2), we can use the formula for the square of a binomial, which states that ((a + b)^2 = a^2 + 2ab + b^2). In our case, (a = 2x) and (b = 5).

Step 1: Identify (a) and (b)

We can set:

• (a = 2x)
• (b = 5)

Step 2: Apply the Binomial Square Formula

Using the binomial square formula, we have:

[
(2x + 5)^2 = (2x)^2 + 2(2x)(5) + (5)^2
]

Step 3: Calculate Each Term

1. Calculate ((2x)^2):
   [
   (2x)^2 = 4x^2
   ]
2. Calculate (2(2x)(5)):
   [
   2(2x)(5) = 20x
   ]
3. Calculate ((5)^2):
   [
   (5)^2 = 25
   ]

Step 4: Combine All Terms

Now we can combine all the calculated terms:

[
(2x + 5)^2 = 4x^2 + 20x + 25
]

Final Result

Thus, the expanded form of ((2x + 5)^2) is:

[
\boxed{4x^2 + 20x + 25}
]

Explanation

In the expansion of a binomial, we are effectively multiplying the binomial by itself. The binomial formula helps us to organize this multiplication systematically. Each term in the binomial is multiplied by every other term, ensuring that we account for all possible products.

Here, squaring the first term ((2x)) gives us the leading term of the polynomial. The middle term arises from the product of the two terms multiplied by two, as we are combining the effects of both terms on each other. The last term is the square of the constant, giving the polynomial a constant component.

This process illustrates the principles of polynomial expansion and showcases how algebraic identities can simplify calculations, making them less prone to error and more efficient.