Question

Write the polynomial x^2+4x-3 as a linear combination of 2x^2-3x and x^2-2x+5 and x+3

Answer 1

Let

$$x^2+4x-3=a(2x^2-3x)+b(x^2-2x+5)+c(x+3)$$

$$x^2+4x-3=(2a+b)x^2+(-3a-2b+c)x+(5b+3c)$$

$$x^2:2a+b=1$$

$$x^1:-3a-2b+c=4$$

$$x^0:5b+3c=-3$$

$$a=\dfrac{1}{2}-\dfrac{1}{2}b$$

$$c=-1-\dfrac{5}{3}b$$

$$-\dfrac{3}{2}+\dfrac{3}{2}b-2b-1-\dfrac{5}{3}b=4$$

$$-\dfrac{13}{6}b=\dfrac{13}{2}$$

$$b=-3$$

$$a=2$$

$$c=4$$

$$x^2+4x-3=2(2x^2-3x)-3(x^2-2x+5)+4(x+3)$$