When i solved this equation i got x=0.5

however when i plugged it back into the original equation i got x=1

what does this mean?

You might have made an error the first time you solved for x. I got x = -0.5.

When you have your log base 4, the way you cancel that out is by making 4 the base on both sides, so you get 4^(log4) to reduce to 1, and you're left with:

2x + 3 = 4^(1/2) ... Simplify

2x + 3 = 2

2x = -1

x = -1/2

If you plug that back in, everything checks out. Maybe double check your use of logarithm/exponent properties?

infinitely many solutions

step-by-step explanation:

we have been given the equation;

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

we need to determine the value of x.

2(2x + 3), we expand this expression to obtain 4x + 6

similarly 2(2x + 5) will become 4x + 10

now we have;

4x + 6 = -4 + 4x + 10

4x + 6 = 4x + 10 - 4

4x + 6 = 4x + 6

the expression on the left hand side is identical to the one on the right hand side. therefore, we are looking at just a single straight line whose equation is;

y = 4x + 6

consequently, we shall have infinitely many solutions