To determine the value of [tex]\( a \)[/tex] in the polynomial resulting from the product [tex]\((y - 4x)(y^2 + 4y + 16)\)[/tex], we will use the distributive property to expand the expression step by step.
1. First, expand the expression using the distributive property:
[tex]\[
(y - 4x)(y^2 + 4y + 16)
\][/tex]
This means we need to distribute each term in the first parenthesis to every term in the second parenthesis.
2. Distribute [tex]\( y \)[/tex]:
[tex]\[
y \cdot (y^2 + 4y + 16) = y \cdot y^2 + y \cdot 4y + y \cdot 16
\][/tex]
[tex]\[
= y^3 + 4y^2 + 16y
\][/tex]
3. Distribute [tex]\(-4x \)[/tex]:
[tex]\[
-4x \cdot (y^2 + 4y + 16) = -4x \cdot y^2 + -4x \cdot 4y + -4x \cdot 16
\][/tex]
[tex]\[
= -4xy^2 - 16xy - 64x
\][/tex]
4. Combine all the terms:
[tex]\[
(y - 4x)(y^2 + 4y + 16) = y^3 + 4y^2 + 16y - 4xy^2 - 16xy - 64x
\][/tex]
5. Group the like terms together to obtain the polynomial in the desired form:
[tex]\[
y^3 + 4y^2 + 16y - 4xy^2 - 16xy - 64x
\][/tex]
Rewriting this in a form that clearly groups the coefficients and like terms:
[tex]\[
y^3 + 4y^2 + ay - 4xy^2 - ax y - 64x
\][/tex]
6. Comparing the polynomial obtained from the expansion with the polynomial [tex]\( y^3 + 4y^2 + ay - 4xy^2 - ax y - 64x \)[/tex], we see that the coefficients of the like terms must match. Specifically, the term involving [tex]\( y \)[/tex] and the term involving [tex]\( xy \)[/tex] need to be analyzed:
- The coefficient of [tex]\( y \)[/tex] in the expanded form is [tex]\( 16y \)[/tex].
- The corresponding term in the form we want is [tex]\( ay \)[/tex].
Hence:
[tex]\[
a = 16
\][/tex]
Therefore, the value of [tex]\( a \)[/tex] in the polynomial is [tex]\( \boxed{16} \)[/tex].
