To factor the quadratic expression \(10a^2 + 13a - 3\), we will use the method of splitting the middle term:

### Step 1: Multiply the coefficient of \(a^2\) (which is 10) by the constant term (which is -3):
\[
10 \times (-3) = -30
\]

### Step 2: Find two numbers that multiply to give -30 and add to give the middle term's coefficient, 13. These numbers are 15 and -2, because:
\[
15 \times (-2) = -30 \quad \text{and} \quad 15 + (-2) = 13
\]

### Step 3: Rewrite the middle term (13a) using 15a and -2a:
\[
10a^2 + 15a - 2a - 3
\]

### Step 4: Factor by grouping:
Group the first two terms and the last two terms:
\[
(10a^2 + 15a) - (2a + 3)
\]

Now, factor each group:
\[
5a(2a + 3) - 1(2a + 3)
\]

### Step 5: Factor out the common binomial factor \((2a + 3)\):
\[
(5a - 1)(2a + 3)
\]

### Final Answer:
The factored form of \(10a^2 + 13a - 3\) is:
\[
(5a - 1)(2a + 3)
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you factor quadratics with negative coefficients?
2. What is the general process for factoring quadratic trinomials?
3. How does factoring help in solving quadratic equations?
4. Can all quadratic expressions be factored? Why or why not?
5. How can you check if your factorization is correct?

**Tip:** Always verify your factorization by expanding the factors to ensure they match the original expression.

Learn how to factor the quadratic expression 10a² + 13a - 3 using the method of splitting the middle term and factoring by grouping. This step-by-step solution covers key concepts in algebra and polynomials, ideal for students in Grades 9-10.

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