To solve the equation \( x^2 + 10x - 40 = 4x \) by factoring, we first rearrange the equation to one side:
\[ x^2 + 10x - 40 - 4x = 0 \]
This simplifies to:
\[ x^2 + 6x - 40 = 0 \]
Next, we want to factor the quadratic equation \( x^2 + 6x - 40 \). We need to find two numbers that multiply to \(-40\) (the constant term) and add to \(6\) (the coefficient of the linear term).
The two numbers that satisfy this condition are \(10\) and \(-4\):
\[ 10 \times (-4) = -40 \quad \text{and} \quad 10 + (-4) = 6 \]
Now, we can factor the quadratic:
\[ (x + 10)(x - 4) = 0 \]
Next, we set each factor equal to zero:
-
\( x + 10 = 0 \) \[ x = -10 \]
-
\( x - 4 = 0 \) \[ x = 4 \]
Thus, the solutions to the equation \( x^2 + 10x - 40 = 4x \) are:
\[ \boxed{-10} \quad \text{and} \quad \boxed{4} \]