To find an expression equivalent to the polynomial [tex]\( 16x^2 + 4 \)[/tex], let's go through the factoring process.
First, let's start from the given polynomial:
[tex]\[ 16x^2 + 4 \][/tex]
Notice that [tex]\( 16x^2 + 4 \)[/tex] can be factored by first identifying the common factor:
[tex]\[ 16x^2 + 4 = 4(4x^2 + 1) \][/tex]
Next, we need to factor [tex]\( 4x^2 + 1 \)[/tex]. This expression is a sum of squares, and can be written as:
[tex]\[ 4x^2 + 1 = (2x)^2 + (1)^2 \][/tex]
The sum of squares can be factored using complex numbers:
[tex]\[ a^2 + b^2 = (a + bi)(a - bi) \][/tex]
Here, [tex]\( a = 2x \)[/tex] and [tex]\( b = 1 \)[/tex]. Substituting these values into the factoring formula, we get:
[tex]\[ 4x^2 + 1 = (2x + i)(2x - i) \][/tex]
Now, substituting this back into our original expression:
[tex]\[ 16x^2 + 4 = 4((2x + i)(2x - i)) \][/tex]
Finally, note that we can further simplify [tex]\( 4((2x + i)(2x - i)) \)[/tex]:
[tex]\[ 4((2x + i)(2x - i)) = (4x + 2i)(4x - 2i) \][/tex]
Therefore, the expression equivalent to the polynomial [tex]\( 16x^2 + 4 \)[/tex] is:
[tex]\[ (4x + 2i)(4x - 2i) \][/tex]
So, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
