Question

question 2 of 5
select the correct answer.
what is the completely factored form of this polynomial?
$2x^4 + 11x^2 + 5$
$\bigcirc\\ (2x + 5)(x + 1)$
$\bigcirc\\ (2x^2 + 5)(x^2 + 1)$
$\bigcirc\\ (2x^2 + 1)(x^2 + 5)$
$\bigcirc\\ (2x + 1)(x + 5)$

Explanation:

Step1: Let \( y = x^2 \), rewrite the polynomial.

The polynomial \( 2x^4 + 11x^2 + 5 \) can be rewritten as \( 2y^2 + 11y + 5 \) by substituting \( y = x^2 \).

Step2: Factor the quadratic in \( y \).

We need to factor \( 2y^2 + 11y + 5 \). We look for two numbers that multiply to \( 2\times5 = 10 \) and add up to \( 11 \). The numbers are \( 10 \) and \( 1 \). So we can rewrite the middle term:
\( 2y^2 + 10y + y + 5 \)
Group the terms:
\( (2y^2 + 10y) + (y + 5) \)
Factor out the common factors from each group:
\( 2y(y + 5) + 1(y + 5) \)
Now factor out \( (y + 5) \):
\( (2y + 1)(y + 5) \)

Step3: Substitute back \( y = x^2 \).

Substituting \( y = x^2 \) back into \( (2y + 1)(y + 5) \), we get \( (2x^2 + 1)(x^2 + 5) \).

Answer:

\( \boldsymbol{(2x^2 + 1)(x^2 + 5)} \) (which corresponds to the option: (2x² + 1)(x² + 5))