We are given the expression: 

\[
-3x^2 + 2x + 8
\]

Let's factorize this quadratic expression step by step.

### Step 1: Multiply the coefficient of \(x^2\) with the constant
Multiply \(-3\) (coefficient of \(x^2\)) by \(8\) (the constant term):
\[
-3 \times 8 = -24
\]

### Step 2: Find factors of \(-24\) that add up to \(2\) (the coefficient of \(x\))
We need two numbers that multiply to \(-24\) and add up to \(2\). The numbers that satisfy this condition are \(6\) and \(-4\):
\[
6 \times (-4) = -24 \quad \text{and} \quad 6 + (-4) = 2
\]

### Step 3: Rewrite the middle term using these factors
Now, rewrite the middle term \(2x\) using the factors \(6x\) and \(-4x\):
\[
-3x^2 + 6x - 4x + 8
\]

### Step 4: Factor by grouping
Group the terms in pairs:
\[
(-3x^2 + 6x) - (4x - 8)
\]

Factor out the common factors from each group:
\[
-3x(x - 2) - 4(x - 2)
\]

### Step 5: Factor out the common binomial factor
Both terms have a common factor \((x - 2)\), so factor it out:
\[
(x - 2)(-3x - 4)
\]

### Final Answer:
The factorized form of the expression \(-3x^2 + 2x + 8\) is:
\[
(x - 2)(-3x - 4)
\]

Would you like more details or have any questions? Here are some related problems:

1. Factorize \( 2x^2 - 5x + 3 \).
2. Solve \( (x - 2)(-3x - 4) = 0 \).
3. Expand \( (x - 2)(-3x - 4) \).
4. Factorize \( -4x^2 + 10x - 6 \).
5. Solve \( x^2 - 6x + 9 = 0 \).

**Tip**: Always check your factors by expanding them back to verify the result.

Learn how to factorize the quadratic expression -3x^2 + 2x + 8 using the grouping method. This detailed step-by-step solution covers important algebraic concepts, including factoring by grouping. Ideal for students in Grades 8-10.

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