To factor the quadratic expression \( 21x^2 + 27x - 30 \), we can use factoring by grouping or another appropriate method such as trial and error or the quadratic formula. Let's go through factoring by grouping.

### Step 1: Multiply \( a \times c \)
For the quadratic equation \( ax^2 + bx + c \), we have:
\[
a = 21, \quad b = 27, \quad c = -30
\]

First, multiply \( a \) and \( c \):
\[
21 \times -30 = -630
\]

### Step 2: Find two numbers that multiply to -630 and add to 27
We are looking for two numbers that multiply to -630 and add to 27.

These numbers are 35 and -18 because:
\[
35 \times -18 = -630
\]
\[
35 + (-18) = 27
\]

### Step 3: Rewrite the middle term using these numbers
We rewrite the middle term \( 27x \) as \( 35x - 18x \):
\[
21x^2 + 35x - 18x - 30
\]

### Step 4: Factor by grouping
Now, group the terms in pairs:
\[
(21x^2 + 35x) - (18x + 30)
\]

Factor out the greatest common factor (GCF) from each group:
\[
7x(3x + 5) - 6(3x + 5)
\]

### Step 5: Factor out the common binomial factor
Both terms contain the common factor \( (3x + 5) \):
\[
(7x - 6)(3x + 5)
\]

### Final Answer
The factored form of \( 21x^2 + 27x - 30 \) is:
\[
(7x - 6)(3x + 5)
\]

Would you like further details or another example?

Here are some related questions:
1. How do you check if a factorization is correct?
2. What is the quadratic formula and how is it used?
3. How do you factor trinomials with larger coefficients?
4. What are common mistakes when factoring?
5. Can all quadratics be factored easily?

**Tip:** Always check your factorization by multiplying the factors back out to ensure you get the original expression.

Learn how to factor the quadratic expression 21x^2 + 27x - 30 using step-by-step methods. This problem covers factoring by grouping, a key algebraic technique. Suitable for high school students learning advanced algebra.

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