To solve the problem, we start by expanding the left-hand side of the given equation and equate coefficients with the right-hand side.

Step 1: Expand the left-hand side

Assume the equation is:
$(x^2 + ax + b)(x + 1) = x^3 + 2x^2 - 5x + c$

Expanding the left-hand side:
$(x^2 + ax + b)(x + 1) = x^3 + x^2 + ax^2 + ax + bx + b$
$= x^3 + (1+a)x^2 + (a+b)x + b$

Step 2: Equate coefficients

Compare with the right-hand side $x^3 + 2x^2 -5x + c$:

• Coefficient of $x^2$: $1+a = 2 \implies a=1$
• Coefficient of $x$: $a+b = -5 \implies 1 + b = -5 \implies b=-6$
• Constant term: $b = c \implies c=-6$

Step 3: Calculate $2a + 3b + c$

$2a + 3b + c = 2(1) + 3(-6) + (-6) = 2 - 18 -6 = -22$

Answer: $\boxed{-22}$

作者声明:本文包含人工智能生成内容。