To factor the polynomial [tex]\( 4a^4 b^2 - 21a^2 b + 5 \)[/tex], let's follow a detailed process to arrive at the correct factored form.
Given the polynomial expression:
[tex]\[
4a^4 b^2 - 21a^2 b + 5
\][/tex]
We need to factor this into two binomials. According to the result obtained, the expression can be factored as follows:
[tex]\[
(4a^2b - 1)(a^2b - 5) = 0
\][/tex]
To ensure this matches the polynomial, let's verify by expanding these choices:
1. Multiply the binomials:
[tex]\[
(4a^2b - 1)(a^2b - 5)
\][/tex]
2. Use distributive property (FOIL method):
[tex]\[
(4a^2b \times a^2b) + (4a^2b \times (-5)) + ((-1) \times a^2b) + ((-1) \times (-5))
\][/tex]
3. Calculate each term:
[tex]\[
4a^4b^2 - 20a^2b - a^2b + 5
\][/tex]
4. Combine like terms:
[tex]\[
4a^4b^2 - 21a^2b + 5
\][/tex]
This matches our original polynomial [tex]\( 4a^4 b^2 - 21a^2 b + 5 \)[/tex].
Thus, the completely factored form of the polynomial [tex]\( 4a^4 b^2 - 21a^2 b + 5 \)[/tex] is:
[tex]\[
(4a^2b - 1)(a^2b - 5) = 0
\][/tex]
So the correct answer is:
A. [tex]\(\left(4a^2b-1\right)\left(a^2b-5\right)=0\)[/tex]
