To solve the expressions \( F \) and \( H \), we will break them down step by step.

### For \( F \):

\[
F = \frac{15}{14} - \left( \frac{17}{23} - \frac{80}{87} + \frac{5}{4} \right) + \left( \frac{17}{23} - \frac{15}{14} + \frac{1}{4} \right)
\]

1. **Calculate \( \frac{17}{23} - \frac{80}{87} + \frac{5}{4} \)**:
- Find a common denominator for the three fractions.
- The least common multiple of \( 23, 87, \) and \( 4 \) is \( 1740 \).
- Convert each fraction:
- \( \frac{17}{23} = \frac{17 \times 60}{23 \times 60} = \frac{1020}{1380} \)
- \( \frac{80}{87} = \frac{80 \times 20}{87 \times 20} = \frac{1600}{1740} \)
- \( \frac{5}{4} = \frac{5 \times 435}{4 \times 435} = \frac{2175}{1740} \)

Now combine:
\[
\frac{1020 - 1600 + 2175}{1740} = \frac{595}{1740}
\]

2. **Calculate \( \frac{17}{23} - \frac{15}{14} + \frac{1}{4} \)**:
- The common denominator will again be \( 1740 \):
- Convert each:
- \( \frac{15}{14} = \frac{15 \times 123}{14 \times 123} = \frac{1845}{1740} \)
- \( \frac{1}{4} = \frac{1 \times 435}{4 \times 435} = \frac{435}{1740} \)

Now combine:
\[
\frac{1020 - 1845 + 435}{1740} = \frac{-390}{1740}
\]

3. **Combine everything**:
\[
F = \frac{15}{14} - \frac{595}{1740} + \frac{-390}{1740}
\]
First, convert \( \frac{15}{14} \) using the common denominator:
\[
\frac{15 \times 123}{14 \times 123} = \frac{1845}{1740}
\]
So:
\[
F = \frac{1845 - 595 - 390}{1740} = \frac{860}{1740} = \frac{43}{87}
\]

### For \( H \):

\[
H = \frac{4}{15} - \frac{23}{28} \left( -\frac{23}{28} + \frac{-11}{15} + \frac{29}{27} - \frac{2}{27} \right)
\]

1. **Calculate the expression inside the parentheses**:
Combine \(-\frac{23}{28}, -\frac{11}{15}, \frac{29}{27}, -\frac{2}{27}\):
- Find a common denominator for these fractions. \( 28, 15, 27 \) has a least common multiple of \( 7560 \).
- Convert each:
- \( -\frac{23}{28} = -\frac{23 \times 270}{28 \times 270} = -\frac{6210}{7560} \)
- \( -\frac{11}{15} = -\frac{11 \times 504}{15 \times 504} = -\frac{5544}{7560} \)
- \( \frac{29}{27} = \frac{29 \times 280}{27 \times 280} = \frac{8120}{7560} \)
- \( -\frac{2}{27} = -\frac{2 \times 280}{27 \times 280} = -\frac{560}{7560} \)

Now combine:
\[
\frac{-6210 - 5544 + 8120 - 560}{7560} = \frac{-1314}{7560}
\]

2. **Now calculate \( H \)**:
\[
H = \frac{4}{15} - \frac{23}{28} \left( \frac{-1314}{7560} \right)
\]

3. **Calculate the multiplication**:
Convert \( \frac{23}{28} \) to have the same denominator:
\[
\frac{23 \times -1314}{28 \times 7560} = \frac{-30222}{211680}
\]

4. **Now combine \( H \)**:
Convert \( \frac{4}{15} \):
\[
\frac{4 \times 14112}{15 \times 14112} = \frac{56448}{211680}
\]
So:
\[
H = \frac{56448 + 30222}{211680} = \frac{86670}{211680} = \frac{4310}{10584}
\]

### Final Results
- \( F = \frac{43}{87} \)
- \( H = \frac{4310}{10584} \) (which can be simplified further if needed, but it's a direct form here).

You may ask for further detail or simplification!