Let's find the sum of the polynomials [tex]\(4x^2 + 4x + 6\)[/tex] and [tex]\(-7x^3 + 2x + 2\)[/tex] step-by-step.
1. Write down the polynomials: We start with these two polynomials:
- The first polynomial is: [tex]\( 4x^2 + 4x + 6 \)[/tex]
- The second polynomial is: [tex]\( -7x^3 + 2x + 2 \)[/tex]
2. Combine like terms:
- For the [tex]\(x^3\)[/tex] terms: The first polynomial has no [tex]\(x^3\)[/tex] term, so we just have the [tex]\(-7x^3\)[/tex] term from the second polynomial.
- For the [tex]\(x^2\)[/tex] terms: There is a [tex]\(4x^2\)[/tex] term in the first polynomial and none in the second polynomial.
- For the [tex]\(x\)[/tex] terms: There is a [tex]\(4x\)[/tex] term in the first polynomial and a [tex]\(2x\)[/tex] term in the second polynomial. Adding these together gives [tex]\(4x + 2x = 6x\)[/tex].
- For the constant terms: There is a [tex]\(6\)[/tex] in the first polynomial and a [tex]\(2\)[/tex] in the second polynomial. Adding these together gives [tex]\(6 + 2 = 8\)[/tex].
3. Combine all the terms: Putting all these combined terms together, we get:
[tex]\[
-7x^3 + 4x^2 + 6x + 8
\][/tex]
Therefore, the sum of the two polynomials [tex]\(\left(4 x^2+4 x+6\right)+\left(-7 x^3+2 x+2\right)\)[/tex] is:
[tex]\[
-7x^3 + 4x^2 + 6x + 8
\][/tex]
