The given equation involves two expressions that we are asked to sum. Let's break down both sides and then combine them step by step.

The expression is:

$\frac{1}{2}a^2 - \frac{3}{4}b^2 + \frac{3}{4}a^2 + \frac{5}{6}b^2 - \frac{2}{5}$

Steps to combine like terms:

1. Group terms with $a^2$:

   • $\frac{1}{2}a^2 + \frac{3}{4}a^2$

   We need a common denominator for these fractions: $\frac{1}{2}a^2 = \frac{2}{4}a^2$ So: $\frac{2}{4}a^2 + \frac{3}{4}a^2 = \frac{5}{4}a^2$

2. Group terms with $b^2$:

   • $-\frac{3}{4}b^2 + \frac{5}{6}b^2$

   Again, we need a common denominator: $\text{LCM of 4 and 6 is 12.}$ Rewriting both fractions with denominator 12: $-\frac{3}{4}b^2 = -\frac{9}{12}b^2, \quad \frac{5}{6}b^2 = \frac{10}{12}b^2$ So: $-\frac{9}{12}b^2 + \frac{10}{12}b^2 = \frac{1}{12}b^2$

3. The constant term remains:

   • $-\frac{2}{5}$

Final expression:

After summing the terms, the final result is:

$\frac{5}{4}a^2 + \frac{1}{12}b^2 - \frac{2}{5}$

Would you like a further breakdown or more details?

Here are 5 related questions for further exploration:

1. How would the result change if one of the constants (like $\frac{2}{5}$) was a variable?
2. Can this expression be simplified further if specific values for $a$ and $b$ are given?
3. What would the steps be if you had to solve for $a$ or $b$ given specific values?
4. How does the result change if different powers of $a$ or $b$ are involved, such as $a^3$ or $b^3$?
5. Can you identify the significance of least common denominators in simplifying expressions?

Tip: When adding algebraic fractions, always make sure to find the least common denominator (LCD) to combine them efficiently!