Sure, let's solve the problem step by step.
Given the expression [tex]\((4a^2 - 6ab + 9b^2)(2a + 3b)\)[/tex] with [tex]\(a = 2\)[/tex] and [tex]\(b = 1\)[/tex]:
1. Evaluate the polynomial inside the parentheses [tex]\((4a^2 - 6ab + 9b^2)\)[/tex]:
[tex]\[
4a^2 - 6ab + 9b^2
\][/tex]
Substituting [tex]\(a = 2\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[
4(2)^2 - 6(2)(1) + 9(1)^2
\][/tex]
Calculate each term:
[tex]\[
4(4) - 6(2) + 9(1)
\][/tex]
[tex]\[
16 - 12 + 9
\][/tex]
[tex]\[
16 - 12 = 4
\][/tex]
[tex]\[
4 + 9 = 13
\][/tex]
So, [tex]\(4a^2 - 6ab + 9b^2 = 13\)[/tex] when [tex]\(a=2\)[/tex] and [tex]\(b=1\)[/tex].
2. Evaluate the linear term [tex]\((2a + 3b)\)[/tex]:
[tex]\[
2a + 3b
\][/tex]
Substituting [tex]\(a = 2\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[
2(2) + 3(1)
\][/tex]
Calculate each term:
[tex]\[
4 + 3
\][/tex]
So, [tex]\(2a + 3b = 7\)[/tex] when [tex]\(a=2\)[/tex] and [tex]\(b=1\)[/tex].
3. Multiply the results from Step 1 and Step 2:
We have:
[tex]\[
(4a^2 - 6ab + 9b^2)(2a + 3b) = (13)(7)
\][/tex]
Calculate the product:
[tex]\[
13 \times 7 = 91
\][/tex]
So, the value of the expression [tex]\((4a^2 - 6ab + 9b^2)(2a + 3b)\)[/tex] when [tex]\(a = 2\)[/tex] and [tex]\(b = 1\)[/tex] is [tex]\(91\)[/tex].
In summary:
- [tex]\(4a^2 - 6ab + 9b^2 = 13\)[/tex]
- [tex]\(2a + 3b = 7\)[/tex]
- The final result is [tex]\(91\)[/tex].
