Question

If T:P1→P1 is a linear transformation such that T(1+2x)=4−2x and T(3+5x)=−3+3x, then T(−4−2x)=

Answer 1

Well, it was tricky on my part, maybe you have a quicker way to do it, but I guess the way to do it is to express (-4-2x) as a linear combination of (1+2x) and (3+5x):
-4-2x = 14(1+2x) - 6(3+5x)
therefore\
T(-4-2x) = T[ 14(1+2x) - 6(3+5x) ]

T is a linear transformation, so it distributes over addition (subtraction) and constants (scalars) may cross the operator:

T(-4-2x) = 14 T(1+2x) - 6 T(3+5x)
= 14( 4-2x ) - 6( -3+3x )
= 74 - 36x

There you go...