Question

use the vertical method to multiply \\((4a^3 - 2a + 3a^2 + 1)\\) and \\((3 - 2a + a^2)\\). what would be the value of a?

Explanation:

Identify the term corresponding to A

The vertical multiplication aligns terms by their degree. The top polynomial is written in standard form:

$$ 4a^3 + 3a^2 - 2a + 1 $$

The second row of partial products is obtained by multiplying the top polynomial by the middle term of the multiplier, which is \(-2a\):

$$ -2a \cdot (4a^3 + 3a^2 - 2a + 1) = -8a^4 - 6a^3 + 4a^2 - 2a $$

Match with the visual layout

Looking at the second row of partial products in the image:

$$ -8a^4 - 6a^3 + A - 2a $$

Comparing this term-by-term with the calculated product:

$$ A = 4a^2 $$

Answer:

\(4a^2\)