Sure! Let's go through the expansion of each notable product step by step and determine the missing terms, coefficients, and signs.
1. Expand [tex]\((x - 2y)^2\)[/tex]:
[tex]\[
(x - 2y)^2 = x^2 + 2(x)(-2y) + (-2y)^2 = x^2 - 4xy + 4y^2
\][/tex]
2. Expand [tex]\((3a + 7)^2\)[/tex]:
[tex]\[
(3a + 7)^2 = (3a)^2 + 2(3a)(7) + 7^2 = 9a^2 + 42a + 49
\][/tex]
3. Expand [tex]\((6x^2 - 1)^2\)[/tex]:
[tex]\[
(6x^2 - 1)^2 = (6x^2)^2 + 2(6x^2)(-1) + (-1)^2 = 36x^4 - 12x^2 + 1
\][/tex]
4. Expand [tex]\((2ab + 5)^2\)[/tex]:
[tex]\[
(2ab + 5)^2 = (2ab)^2 + 2(2ab)(5) + 5^2 = 4a^2b^2 + 20ab + 25
\][/tex]
5. Expand [tex]\((3r^2t - 2)^2\)[/tex]:
[tex]\[
(3r^2t - 2)^2 = (3r^2t)^2 + 2(3r^2t)(-2) + (-2)^2 = 9r^4t^2 - 12r^2t + 4
\][/tex]
Now, for the list of missing terms (sign and coefficients except in the first term):
[tex]\[
x^2 \boxed{-4} \boxed{xy} \boxed{+4} \boxed{y^2}
\][/tex]
[tex]\[
a^2 \boxed{+42} \boxed{a} \boxed{+49}
\][/tex]
[tex]\[
\boxed{36} \boxed{x^4} \boxed{-12} \boxed{x^2} \boxed{+1}
\][/tex]
[tex]\[
\boxed{4} \boxed{a^2} \boxed{b^2} \boxed{+20} \boxed{ab} \boxed{+25}
\][/tex]
[tex]\[
\boxed{9} \boxed{r^4} \boxed{t^2} \boxed{-12} \boxed{r^2} \boxed{t} \boxed{+4}
\][/tex]
Putting the completed forms all together:
1. [tex]\((x-2y)^2 = x^2 \boxed{-4} \boxed{xy} \boxed{+4} \boxed{y^2} \)[/tex]
2. [tex]\((3a+7)^2 = a^2 \boxed{+42} \boxed{a} \boxed{+49} \)[/tex]
3. [tex]\((6x^2-1)^2 = \boxed{36} \boxed{x^4} \boxed{-12} \boxed{x^2} \boxed{+1} \)[/tex]
4. [tex]\((2ab+5)^2 = \boxed{4} \boxed{a^2} \boxed{b^2} \boxed{+20} \boxed{ab} \boxed{+25} \)[/tex]
5. [tex]\((3r^2t-2)^2 = \boxed{9} \boxed{r^4} \boxed{t^2} \boxed{-12} \boxed{r^2} \boxed{t} \boxed{+4} \)[/tex]
I hope this detailed solution helps you understand how to complete the notable products!
