Question: Factor the trinomial. \[ a^4 + 46a^2 + 45 \] Select the correct choice below and, if necessary,…

Factor the trinomial.

\[ a^4 + 46a^2 + 45 \]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. \( a^4 + 46a^2 + 45 = \) (Factor completely.)

B. The polynomial is prime.

Solution

To factor the trinomial \( a^4 + 46a^2 + 45 \), we can treat it as a quadratic in terms of \( a^2 \). Let \( x = a^2 \). Then the trinomial becomes \( x^2 + 46x + 45 \). We need to find two numbers that multiply to 45 and add up to 46. The numbers that satisfy this condition are 1 and 45. Thus, the trinomial factors as: \[ (x + 1)(x + 45) \] Now, substitute back \( x = a^2 \): \[ (a^2 + 1)(a^2 + 45) \] Therefore, the factored form of \( a^4 + 46a^2 + 45 \) is: \[ (a^2 + 1)(a^2 + 45) \]