Expand Factor 1E2D44

Question: User: 1. If $ax^2 + bx + c = (2x - 3)(3x + 4)$ for all values of $x$, find the values of $a$, $b$ and $c$. 2. If $(3x - 2)(x + 5) = 3x^2 + px + q$ for all values of $x$, find the values of $p$ and $q$. 3. If $x^2 + 6x + 16 = (x + a)^2 + b$ for all values of $x$, find the values of $a$ and $b$. Find the real numbers $a$ and $b$ such that $x^2 + 4x - 6 = (x + a)^2 + b$ for all $x \in \mathbb{R}$. 5. If $2x^2 + 5x + 6 = p(x + q)^2 + r$ for all values of $x$, find the values of $p$, $q$ and $r$. 6. If $x^2 - 4x - 5 = (x - n)^2 - m$ for all $x$, find the values of $m$ and $n$. 8. The volume of this closed box, $V$, is given by the function $V(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$ and $d \in \mathbb{Z}$. (i) Find the values for $a$, $b$, $c$ and $d$. The external surface area, $S(x)$, is given by the equation $S(x) = px^2 + qx + r$, where $p$, $q$ and $r \in \mathbb{Z}$. (ii) Find the values of $p$, $q$ and $r$. 9. If $3(x - p)^2 + q = 3x^2 - 12x + 7$ for all $x$, find the values of $p$ and $q$. 10. The volume of a solid box is given by $V(x) = x^3 + 12x^2 + bx + 30$. If the top of the box has an area of $x^2 + cx + 4$ and the height is $x + a$, find the values of $a$, $b$ and $c$.