Unsure of what strategy to use.
x^2 + bx + c does not work with fractions.
please take a look at my working out.. I am only focusing on the left side in this case
This is not a factoring problem at all!
What you are doing here is
completing the square to solve the equation [imath]x^2-\frac{3}{2}x=10[/imath].
What you do here is to set things up so that you
already know what the "factored" form (the perfect square) will be. You don't need to look at [imath]x^2-\frac{3}{2}x+\frac{9}{16}[/imath] and factor it.
Here is how I think of this process: Given [imath]x^2-\frac{3}{2}x=10[/imath], I say, "The first two terms look like the result of squaring a binomial, [imath](x+h)^2 = x^2+2hx+h^2[/imath]. What will h be? Since the second term, [imath]-\frac{3}{2}x[/imath], has to equal [imath]2hx[/imath], our h must be half of [imath]-\frac{3}{2}[/imath], which is [imath]-\frac{3}{4}[/imath]. So I want my next line to be [imath]\left(x-\frac{3}{4}\right)^2[/imath].
So the work involved is not to figure out this side by factoring the line above it; it's to add the right term to both sides of the first line so that I will get this on the second.
You're right that the usual method for factoring a trinomial doesn't work well with fractional coefficients. That's why we don't use it.