Question: How to solve 9a and explain why
1. **State the problem:** Factor the quadratic expression $$x^2 + \frac{5}{4}x + \frac{3}{8}$$ completely.
2. **Rewrite the expression to clear fractions:** Multiply the entire expression by 8 (the least common multiple of denominators 4 and 8) to avoid fractions:
$$8 \times \left(x^2 + \frac{5}{4}x + \frac{3}{8}\right) = 8x^2 + 10x + 3$$
3. **Factor the quadratic $$8x^2 + 10x + 3$$:**
We look for two numbers that multiply to $$8 \times 3 = 24$$ and add to $$10$$. These numbers are $$6$$ and $$4$$.
4. **Rewrite the middle term using these numbers:**
$$8x^2 + 6x + 4x + 3$$
5. **Group terms and factor each group:**
$$ (8x^2 + 6x) + (4x + 3) $$
Factor out $$2x$$ from the first group and $$1$$ from the second:
$$ 2x(4x + 3) + 1(4x + 3) $$
6. **Factor out the common binomial:**
$$ (4x + 3)(2x + 1) $$
7. **Remember to divide back by 8 to compensate for the initial multiplication:**
$$ x^2 + \frac{5}{4}x + \frac{3}{8} = \frac{1}{8}(4x + 3)(2x + 1) $$
**Explanation:**
- Multiplying by 8 clears fractions, making factoring easier.
- Factoring by grouping helps break down the middle term.
- Factoring out the greatest common binomial simplifies the expression.
- Dividing by 8 at the end returns to the original expression's scale.
**Final answer:**
$$\boxed{\frac{1}{8}(4x + 3)(2x + 1)}$$