Certainly! Let's work through each part step-by-step.
### Step 1: Evaluating [tex]\((h + k)(2)\)[/tex]
We need to find [tex]\( h(2) \)[/tex] and [tex]\( k(2) \)[/tex], and then add them together.
1. Evaluate [tex]\( h(2) \)[/tex]:
[tex]\[ h(2) = 2^2 + 1 = 4 + 1 = 5 \][/tex]
2. Evaluate [tex]\( k(2) \)[/tex]:
[tex]\[ k(2) = 2 - 2 = 0 \][/tex]
Now, sum these values:
[tex]\[ (h + k)(2) = h(2) + k(2) = 5 + 0 = 5 \][/tex]
So, the answer for [tex]\((h + k)(2)\)[/tex] is:
[tex]\[ \boxed{5} \][/tex]
### Step 2: Evaluating [tex]\((h - k)(3)\)[/tex]
We need to find [tex]\( h(3) \)[/tex] and [tex]\( k(3) \)[/tex], and then subtract [tex]\( k(3) \)[/tex] from [tex]\( h(3) \)[/tex].
1. Evaluate [tex]\( h(3) \)[/tex]:
[tex]\[ h(3) = 3^2 + 1 = 9 + 1 = 10 \][/tex]
2. Evaluate [tex]\( k(3) \)[/tex]:
[tex]\[ k(3) = 3 - 2 = 1 \][/tex]
Now, subtract these values:
[tex]\[ (h - k)(3) = h(3) - k(3) = 10 - 1 = 9 \][/tex]
So, the answer for [tex]\((h - k)(3)\)[/tex] is:
[tex]\[ \boxed{9} \][/tex]
### Step 3: Evaluating [tex]\( 3h(2) + 2k(3) \)[/tex]
We already have [tex]\( h(2) \)[/tex] and [tex]\( k(3) \)[/tex] from the previous steps, which are [tex]\( 5 \)[/tex] and [tex]\( 1 \)[/tex] respectively.
1. Multiply [tex]\( h(2) \)[/tex] by 3:
[tex]\[ 3h(2) = 3 \times 5 = 15 \][/tex]
2. Multiply [tex]\( k(3) \)[/tex] by 2:
[tex]\[ 2k(3) = 2 \times 1 = 2 \][/tex]
Now, add these values:
[tex]\[ 3h(2) + 2k(3) = 15 + 2 = 17 \][/tex]
So, the answer for [tex]\( 3h(2) + 2k(3) \)[/tex] is:
[tex]\[ \boxed{17} \][/tex]
### Summary
Let's summarize all the answers we have found:
[tex]\[
\begin{array}{l}
(h+k)(2) = \boxed{5} \\
(h-k)(3) = \boxed{9} \\
3h(2) + 2k(3) = \boxed{17}
\end{array}
\][/tex]
These detailed steps show how we arrived at these results for each expression.
