$$ 4a-\left(2a^2+3z\right)+\frac45\left(a^2-2a+8\right)\text{ simplify} $$

To simplify the given expression:

\[
4a - \left(2a^2 + 3z\right) + \frac{4}{5} \left(a^2 - 2a + 8\right)
\]

### Step 1: Distribute and Expand

Distribute the negative sign and the fraction inside their respective parentheses:

\[
4a - 2a^2 - 3z + \frac{4}{5}a^2 - \frac{8}{5}a + \frac{32}{5}
\]

### Step 2: Combine Like Terms

Now, let's combine like terms:

- **$a^2$ terms:** $-2a^2 + \frac{4}{5}a^2$
- **$a$ terms:** $4a - \frac{8}{5}a$
- **Constant terms:** $ -3z $ (there's only one term with $z$)
- **Constant terms:** $\frac{32}{5}$

#### Simplifying the Coefficients

1. **Combine the $a^2$ terms:**

\[
-2a^2 + \frac{4}{5}a^2 = \left(-\frac{10}{5} + \frac{4}{5}\right)a^2 = -\frac{6}{5}a^2
\]

2. **Combine the $a$ terms:**

\[
4a - \frac{8}{5}a = \left(\frac{20}{5} - \frac{8}{5}\right)a = \frac{12}{5}a
\]

3. **Include the remaining terms:**

The terms with $z$ and the constant remain the same: $-3z + \frac{32}{5}$

### Step 3: Write the Simplified Expression

Putting all the simplified terms together:

\[
-\frac{6}{5}a^2 + \frac{12}{5}a - 3z + \frac{32}{5}
\]

### Final Simplified Expression:

\[
\boxed{-\frac{6}{5}a^2 + \frac{12}{5}a - 3z + \frac{32}{5}}
\]

Would you like more details or have any other questions?

### Related Questions:

1. How can we further simplify expressions with fractions?
2. What are some strategies for combining like terms effectively?
3. How can we apply the distributive property to more complex expressions?
4. How can we simplify expressions involving multiple variables?
5. How do we handle negative signs in complex expressions?

### Tip:

When simplifying expressions with fractions, it's often helpful to convert all coefficients to a common denominator first.

$$ 4a - \left(2a^2 + 3z\right) + \frac{4}{5} \left(a^2 - 2a + 8\right) \text{ simplify} $$

This page explains how to simplify the algebraic expression 4a - (2a^2 + 3z) + (4/5)(a^2 - 2a + 8) using the distributive property and combining like terms. Suitable for students in Grades 9-10, it covers key algebra concepts and includes a detailed step-by-step solution.

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