To factor the expression \( 2b^2 - 18b + 40 \) completely, we can follow these steps:
-
Factor out the greatest common factor (GCF): First, we notice that there is a common factor of 2 in each term:
\[ 2b^2 - 18b + 40 = 2(b^2 - 9b + 20) \]
-
Factor the quadratic expression: Next, we will factor the quadratic \( b^2 - 9b + 20 \). We need to find two numbers that multiply to 20 (the constant term) and add up to -9 (the coefficient of \( b \)).
The numbers that work are -4 and -5, since:
\[ -4 \times -5 = 20 \quad \text{and} \quad -4 + (-5) = -9 \]
So we can factor the quadratic:
\[ b^2 - 9b + 20 = (b - 4)(b - 5) \]
-
Combine with the GCF: Now, substitute this factorization back into our expression:
\[ 2(b^2 - 9b + 20) = 2(b - 4)(b - 5) \]
Thus, the completely factored form of the expression \( 2b^2 - 18b + 40 \) is:
\[ \boxed{2(b - 4)(b - 5)} \]