To solve the problem and find the simplified form of the expression [tex]\(P + 2Q + 2R\)[/tex] where [tex]\(P\)[/tex], [tex]\(Q\)[/tex], and [tex]\(R\)[/tex] are given, follow these steps:
1. Write down the given polynomials:
[tex]\[
P = 2y^2 - 5 - 2y
\][/tex]
[tex]\[
Q = y^2 - 3y + 6
\][/tex]
[tex]\[
R = y^2 - 4y + 2
\][/tex]
2. Express the expression [tex]\(P + 2Q + 2R\)[/tex]:
[tex]\[
P + 2Q + 2R = (2y^2 - 5 - 2y) + 2(y^2 - 3y + 6) + 2(y^2 - 4y + 2)
\][/tex]
3. Expand and distribute the multipliers [tex]\(2\)[/tex] for [tex]\(Q\)[/tex] and [tex]\(R\)[/tex]:
[tex]\[
2Q = 2(y^2 - 3y + 6) = 2y^2 - 6y + 12
\][/tex]
[tex]\[
2R = 2(y^2 - 4y + 2) = 2y^2 - 8y + 4
\][/tex]
4. Combine all the terms:
[tex]\[
P + 2Q + 2R = (2y^2 - 5 - 2y) + (2y^2 - 6y + 12) + (2y^2 - 8y + 4)
\][/tex]
5. Group like terms together:
Combine all the [tex]\( y^2 \)[/tex], [tex]\( y \)[/tex], and constant terms separately:
[tex]\[
2y^2 + 2y^2 + 2y^2 + (-2y - 6y - 8y) + (-5 + 12 + 4)
\][/tex]
6. Simplify the combined terms:
[tex]\[
6y^2 + (-16y) + 11
\][/tex]
So, the expression [tex]\(P + 2Q + 2R\)[/tex] simplifies to:
[tex]\[
6y^2 - 16y + 11
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[
P + 2Q + 2R = 6y^2 - 16y + 11
\][/tex]
This is the solution, and all the steps show the detailed process of arriving at the simplified expression.
