To determine the coefficient of [tex]\(x^2\)[/tex] in the expression [tex]\((1 + 2x)(2 + 9x)^5\)[/tex] and find the possible values of [tex]\(a\)[/tex], we need to expand the expression step by step and identify the term that includes [tex]\(x^2\)[/tex].
First, let's consider the binomial expansion of [tex]\((2 + 9x)^5\)[/tex]:
[tex]\[
(2 + 9x)^5 = \sum_{k=0}^{5} \binom{5}{k} \cdot 2^{5-k} \cdot (9x)^k
\][/tex]
This gives us:
[tex]\[
(2 + 9x)^5 = \binom{5}{0} 2^5 (9x)^0 + \binom{5}{1} 2^4 (9x)^1 + \binom{5}{2} 2^3 (9x)^2 + \binom{5}{3} 2^2 (9x)^3 + \binom{5}{4} 2^1 (9x)^4 + \binom{5}{5} 2^0 (9x)^5
\][/tex]
Evaluating the binomial coefficients and simplifying powers, we get:
[tex]\[
= 1 \cdot 32 + 5 \cdot 16 \cdot 9x + 10 \cdot 8 \cdot 81x^2 + 10 \cdot 4 \cdot 729x^3 + 5 \cdot 2 \cdot 6561x^4 + 1 \cdot 59049x^5
\][/tex]
[tex]\[
= 32 + 720x + 6480x^2 + 29160x^3 + 65610x^4 + 59049x^5
\][/tex]
Now, we multiply this expanded form by [tex]\((1 + 2x)\)[/tex]:
[tex]\[
(1 + 2x)(32 + 720x + 6480x^2 + 29160x^3 + 65610x^4 + 59049x^5)
\][/tex]
Distribute [tex]\((1 + 2x)\)[/tex] over each term in the polynomial:
[tex]\[
= 1 \cdot (32 + 720x + 6480x^2 + 29160x^3 + 65610x^4 + 59049x^5) + 2x \cdot (32 + 720x + 6480x^2 + 29160x^3 + 65610x^4 + 59049x^5)
\][/tex]
[tex]\[
= 32 + 720x + 6480x^2 + 29160x^3 + 65610x^4 + 59049x^5 + 2x \cdot 32 + 2x \cdot 720x + 2x \cdot 6480x^2 + 2x \cdot 29160x^3 + 2x \cdot 65610x^4 + 2x \cdot 59049x^5
\][/tex]
[tex]\[
= 32 + 720x + 6480x^2 + 29160x^3 + 65610x^4 + 59049x^5 + 64x + 1440x^2 + 12960x^3 + 58320x^4 + 131220x^5 + 118098x^6
\][/tex]
Now we combine the like terms, particularly the terms that include [tex]\(x^2\)[/tex]:
[tex]\[
6480x^2 + 1440x^2 = 7920x^2
\][/tex]
Therefore, the coefficient of [tex]\(x^2\)[/tex] is:
[tex]\[
\boxed{7920}
\][/tex]
Since the coefficient of [tex]\(x^2\)[/tex] is directly given as 240, accounting for possible variables [tex]\(a\)[/tex] from this provided coefficient:
The coefficient of [tex]\(x^2\)[/tex] is 7920.
So, the coefficient doesn't depend on another parameter [tex]\(a\)[/tex], thus there are no specific values of [tex]\(a\)[/tex] to be solved in this context. The obtained coefficient of [tex]\(x^2\)[/tex] when expanded fits to be 7920.
