Comparing coefficients

Coefficients of a polynomial identity ●○○○○ 10 ptsIf $(x+a)(x+8)=x^{2}+bx+24$ for all values of $x$ , find the value of $a+b$ .
Coefficients of a re-written quadratic ●○○○○ 10 ptsThe expression $2(x-3)^{2}-12$ can be re-written as $ax^{2}+bx+c$ for some numbers $a, b, c$ . What is the value of $10a-b-4c$ ?
Coefficients of an algebraic identity ●○○○○ 10 ptsThe equation $ax+18=2(x-b)$ holds for all values of $x$ . Work out the value of $a-b$ .
Quadratic constraint on a shifted expression ●●○○○ 20 ptsGiven $6x^{2}-4x-3=0$ , find the value of $(x-1)^{2}+x\left(x+\frac{2}{3}\right)$ .
Sum of coefficients in a shifted cubic expansion ●●○○○ 20 ptsIf the equation $1+x+x^{2}+x^{3}=a_{0}+a_{1}(1-x)+a_{2}(1-x)^{2}+a_{3}(1-x)^{3}$ holds for all real $x$ , where $a_{0}$ , $a_{1}$ , $a_{2}$ , $a_{3}$ are real constants, find the value of $a_{1}+a_{2}+a_{3}$ .
Quadratic expression in shifted variables ●●○○○ 20 ptsGiven $a=2009x+2007$ , $b=2009x+2008$ , $c=2009x+2009$ , find the value of $a^{2}+b^{2}+c^{2}-ab-bc-ca$ .
Possible values of c in a polynomial identity ●●●○○ 50 ptsGiven integers $a$ , $b$ and $c$ such that the equation $(x+a)(x+b)+c(x-10)=(x-11)(x+1)$ holds for all $x$ , find all possible values of $c$ .

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