To factor the quadratic equation \( 5x^2 - 32x - 21 = 0 \), we'll follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \), where:
- \( a = 5 \)
- \( b = -32 \)
- \( c = -21 \)

### Step 2: Multiply \( a \) and \( c \)
Multiply the leading coefficient \( a = 5 \) by the constant term \( c = -21 \):
\[
a \cdot c = 5 \times (-21) = -105
\]

### Step 3: Find two numbers that multiply to \( -105 \) and add up to \( -32 \)
We need two numbers that multiply to \( -105 \) and add to \( -32 \). These numbers are \( -35 \) and \( 3 \), because:
\[
-35 \times 3 = -105 \quad \text{and} \quad -35 + 3 = -32
\]

### Step 4: Rewrite the middle term
Rewrite \( -32x \) as \( -35x + 3x \):
\[
5x^2 - 35x + 3x - 21 = 0
\]

### Step 5: Factor by grouping
Group the terms:
\[
(5x^2 - 35x) + (3x - 21) = 0
\]
Factor out the greatest common factor from each group:
\[
5x(x - 7) + 3(x - 7) = 0
\]

### Step 6: Factor out the common binomial factor
Factor out \( (x - 7) \):
\[
(5x + 3)(x - 7) = 0
\]

### Step 7: Solve for \( x \)
Now, set each factor equal to zero and solve for \( x \):
\[
5x + 3 = 0 \quad \text{or} \quad x - 7 = 0
\]
Solving these:
\[
x = -\frac{3}{5} \quad \text{or} \quad x = 7
\]

So, the factored form of the quadratic equation is:
\[
(5x + 3)(x - 7) = 0
\]

And the solutions are:
\[
x = -\frac{3}{5}, \quad x = 7
\]

Would you like more details or have any questions?

### Related Questions:
1. How can I check if my factored form is correct?
2. What if the equation had complex roots? How would factoring change?
3. Can every quadratic equation be factored?
4. What is the difference between factoring and using the quadratic formula?
5. How does the discriminant help determine whether a quadratic is factorable?

**Tip:** When factoring quadratics, always start by checking for common factors across all terms before applying other methods.

Learn how to factor and solve the quadratic equation 5x^2 − 32x − 21 with this detailed step-by-step solution. This problem explains factoring by grouping and applying the zero-product property. Perfect for students in Grades 9-11.

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