To solve this problem, we will follow the mathematical steps required for polynomial multiplication using the Vertical Method and compare the resulting product's coefficients to find the value of [tex]\( B \)[/tex].
### Step-by-Step Solution
1. Multiply the Polynomials:
[tex]\[
P_1(a) = 4a^3 + 3a^2 - 2a + 1
\][/tex]
[tex]\[
P_2(a) = 3 - 2a + a^2
\][/tex]
Using the Vertical Method, we multiply each term in [tex]\( P_1 \)[/tex] by each term in [tex]\( P_2 \)[/tex]:
[tex]\[
\begin{array}{r}
\quad\quad\quad\quad 4a^3 + 3a^2 - 2a + 1 \\
\times \quad\quad\quad\quad (3 - 2a + a^2) \\
\hline
4a^3 \cdot 3 + 3a^2 \cdot 3 - 2a \cdot 3 + 1 \cdot 3 \\
4a^3 \cdot (-2a) + 3a^2 \cdot (-2a) - 2a \cdot (-2a) + 1 \cdot (-2a) \\
4a^3 \cdot a^2 + 3a^2 \cdot a^2 - 2a \cdot a^2 + 1 \cdot a^2 \\
\end{array}
\][/tex]
After performing these multiplications, we sum the results:
[tex]\[
\begin{array}{r}
(4a^3 \cdot 3) + (3a^2 \cdot 3) + (-2a \cdot 3) + (1 \cdot 3) \\
= 12a^3 + 9a^2 - 6a + 3 \\
\end{array}
\][/tex]
[tex]\[
\begin{array}{r}
(4a^3 \cdot -2a) + (3a^2 \cdot -2a) + (-2a \cdot -2a) + (1 \cdot -2a) \\
= -8a^4 - 6a^3 + 4a^2 - 2a \\
\end{array}
\][/tex]
[tex]\[
\begin{array}{r}
(4a^3 \cdot a^2) + (3a^2 \cdot a^2) + (-2a \cdot a^2) + (1 \cdot a^2) \\
= 4a^5 + 3a^4 - 2a^3 + a^2 \\
\end{array}
\][/tex]
Combining all the results:
[tex]\[
4a^5 + 3a^4 - 8a^4 + 3a^4 - 6a^3 + 12a^3 - 2a^3 - 6a + 4a^2 + a^2 + 3 = 4a^5 - 5a^4 + 4a^3 + 14a^2 - 8a + 3
\][/tex]
2. Compare Polynomials:
Given that [tex]\(\frac{4a^5 + 3a^4 - 2a^3 + a^2}{4a^5 - 5a^4 + B + 14a^2 - 8a + 3}\)[/tex],
we need to find [tex]\( B \)[/tex] such that the denominator matches the product obtained from the multiplication.
By comparing the product obtained from the multiplication and the given expression, we have:
[tex]\( \text{Numerator: } 4a^5 + 3a^4 - 2a^3 + a^2 \)[/tex]
[tex]\( \text{Denominator: } 4a^5 - 5a^4 + B + 14a^2 - 8a + 3 \)[/tex]
3. Identifying [tex]\( B \)[/tex]:
By comparing the terms in the polynomial product and given denominator, we observe:
[tex]\[
4a^5, -5a^4, \text{and so on, match the terms exactly, up to } 14a^2 - 8a + 3
\][/tex]
To find [tex]\( B \)[/tex]:
[tex]\( \text{Comparing the coefficient of } a^4: 3a^4 - (-5a^4) = 4a \)[/tex]
Hence, [tex]\( B = -8 \)[/tex].
Thus, the value of [tex]\( A \)[/tex] is [tex]\(\boxed{-8}\)[/tex].
