Sure! Let's break down the problem and simplify the given expression step-by-step.
We are given the expression:
[tex]\[
\frac{3}{4}(2h + 9) - \frac{5}{4}(h - 1) + \frac{H + 32}{4}
\][/tex]
First, let's simplify each part of the expression individually.
### Part 1: [tex]\(\frac{3}{4}(2h + 9)\)[/tex]
Expand the expression by distributing [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\frac{3}{4} \cdot 2h + \frac{3}{4} \cdot 9 = \frac{3 \cdot 2h}{4} + \frac{3 \cdot 9}{4} = \frac{6h}{4} + \frac{27}{4}
\][/tex]
Simplify the fractions:
[tex]\[
\frac{6h}{4} = \frac{3h}{2}
\][/tex]
Thus, the simplified form of Part 1 is:
[tex]\[
\frac{3h}{2} + \frac{27}{4}
\][/tex]
### Part 2: [tex]\(\frac{5}{4}(h - 1)\)[/tex]
Expand the expression by distributing [tex]\(\frac{5}{4}\)[/tex]:
[tex]\[
\frac{5}{4} \cdot h - \frac{5}{4} \cdot 1 = \frac{5h}{4} - \frac{5}{4}
\][/tex]
So, the simplified form of Part 2 is:
[tex]\[
\frac{5h}{4} - \frac{5}{4}
\][/tex]
### Part 3: [tex]\(\frac{H + 32}{4}\)[/tex]
This part is already in its simplest form; no distribution needed here.
### Combining the simplified parts
Now we combine all the simplified parts:
[tex]\[
\left( \frac{3h}{2} + \frac{27}{4} \right) - \left( \frac{5h}{4} - \frac{5}{4} \right) + \frac{H + 32}{4}
\][/tex]
Distribute the subtraction over the second set of terms:
[tex]\[
\frac{3h}{2} + \frac{27}{4} - \frac{5h}{4} + \frac{5}{4} + \frac{H + 32}{4}
\][/tex]
Next, let's combine the like terms. We'll group the terms with [tex]\(h\)[/tex] together and the constants together.
First, convert [tex]\(\frac{3h}{2}\)[/tex] to a common denominator of 4 to combine with [tex]\(\frac{5h}{4}\)[/tex]:
[tex]\[
\frac{3h}{2} = \frac{3h \cdot 2}{2 \cdot 2} = \frac{6h}{4}
\][/tex]
Now we can combine the [tex]\(h\)[/tex] terms:
[tex]\[
\frac{6h}{4} - \frac{5h}{4} = \frac{(6h - 5h)}{4} = \frac{h}{4}
\][/tex]
Now combine the constants:
[tex]\[
\frac{27}{4} + \frac{5}{4} + \frac{H + 32}{4}
\][/tex]
Combine the constants with the common denominator:
[tex]\[
\frac{(27 + 5) + (H + 32)}{4} = \frac{32 + H + 32}{4} = \frac{H + 64}{4}
\][/tex]
So, the complete simplified expression is:
[tex]\[
\frac{h}{4} + \frac{H + 64}{4}
\][/tex]
Combine the terms under the common denominator:
[tex]\[
\frac{h + H + 64}{4}
\][/tex]
Thus, the simplified form of the entire expression is:
[tex]\[
\boxed{\frac{h + H + 64}{4}}
\][/tex]
