Sure! Let's solve the problem step-by-step.
We are asked to subtract the polynomial [tex]\(7x^2 - x - 1\)[/tex] from the polynomial [tex]\(x^2 + 3x + 3\)[/tex].
1. Write down the two polynomials:
[tex]\[
p_1(x) = x^2 + 3x + 3
\][/tex]
[tex]\[
p_2(x) = 7x^2 - x - 1
\][/tex]
2. Perform the subtraction:
[tex]\[
p_1(x) - p_2(x) = (x^2 + 3x + 3) - (7x^2 - x - 1)
\][/tex]
3. Distribute the negative sign through the second polynomial:
[tex]\[
(x^2 + 3x + 3) - 7x^2 + x + 1
\][/tex]
4. Combine like terms:
- For the [tex]\(x^2\)[/tex] terms:
[tex]\[
x^2 - 7x^2 = -6x^2
\][/tex]
- For the [tex]\(x\)[/tex] terms:
[tex]\[
3x + x = 4x
\][/tex]
- For the constant terms:
[tex]\[
3 + 1 = 4
\][/tex]
5. Put it all together:
[tex]\[
-6x^2 + 4x + 4
\][/tex]
Therefore, the result of subtracting the polynomial [tex]\(7x^2 - x - 1\)[/tex] from the polynomial [tex]\(x^2 + 3x + 3\)[/tex] is:
[tex]\[
-6x^2 + 4x + 4
\][/tex]
This is the final simplified result.
