To determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that [tex]\((x+2)\)[/tex] and [tex]\((x-1)\)[/tex] are factors of the quadratic expression [tex]\( x^2 + (a+2)x + a + b \)[/tex], we start by considering what it means for these factors to be true.
Given that [tex]\((x+2)\)[/tex] and [tex]\((x-1)\)[/tex] are factors, we can express the quadratic equation as:
[tex]\[
(x+2)(x-1)
\][/tex]
Next, we expand [tex]\((x+2)(x-1)\)[/tex]:
[tex]\[
(x + 2)(x - 1) = x(x - 1) + 2(x - 1) = x^2 - x + 2x - 2 = x^2 + x - 2
\][/tex]
Now, this expanded form should be identical to the original quadratic expression [tex]\( x^2 + (a+2)x + a + b \)[/tex]:
[tex]\[
x^2 + (a+2)x + a + b = x^2 + x - 2
\][/tex]
To find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we compare the coefficients of corresponding terms:
For the [tex]\( x \)[/tex]-coefficients:
[tex]\[
a + 2 = 1
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a + 2 = 1 \implies a = 1 - 2 \implies a = -1
\][/tex]
For the constant terms:
[tex]\[
a + b = -2
\][/tex]
We already found [tex]\( a = -1 \)[/tex], so substitute [tex]\( a \)[/tex] into the equation:
[tex]\[
-1 + b = -2 \implies b = -2 + 1 \implies b = -1
\][/tex]
Thus, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are [tex]\( -1 \)[/tex] and [tex]\( -1 \)[/tex] respectively.
Therefore, the correct answer is:
[tex]\[
\boxed{-1 \quad -1}
\][/tex]
