To solve the given division problem [tex]\(\frac{{x^2 + x}}{{2x + 2}}\)[/tex], we can follow these steps:
1. Simplify the denominator: Notice that the denominator [tex]\(2x + 2\)[/tex] can be factored as [tex]\(2(x + 1)\)[/tex].
2. Rewrite the expression: Now, we can rewrite the division as:
[tex]\[
\frac{{x^2 + x}}{{2(x + 1)}}
\][/tex]
3. Perform polynomial long division:
- Divide the leading term of the numerator [tex]\(x^2\)[/tex] by the leading term of the denominator [tex]\(2x\)[/tex]. This gives us [tex]\(\frac{x^2}{2x} = \frac{x}{2}\)[/tex].
- Multiply [tex]\(\frac{x}{2}\)[/tex] by the entire denominator [tex]\(2(x + 1)\)[/tex], which results in [tex]\(\frac{x}{2} \cdot 2(x + 1) = x(x + 1) = x^2 + x\)[/tex].
- Subtract this result from the original numerator:
[tex]\[
(x^2 + x) - (x^2 + x) = 0
\][/tex]
4. Quotient and remainder: The quotient from the division is [tex]\(\frac{x}{2}\)[/tex] and the remainder is [tex]\(0\)[/tex].
Therefore, the quotient of the division problem [tex]\(\frac{x^2 + x}{2x + 2}\)[/tex] is [tex]\(\boxed{\frac{x}{2}}\)[/tex].
