J.M.'s advice, given in his comment, is one solution. Here is another using the plot option Evaluated. I also also add custom x-axis ticks and give the option Appearance -> "Labeled"Appearance -> "Labeled" to the control variable a to make the manipulation more user friendly.
Manipulate[ Plot[FourierTrigSeries[-x^2 + a x, x, 10], {x, -2 Pi, 2 Pi}, Ticks -> {Subdivide[-2 Pi, 2 Pi, 8], Automatic}, Evaluated -> True], {a, -10, 10, .2, Appearance -> "Labeled"}] 
J.M.'s advice, given in his comment, is one solution. Here is another using the plot option Evaluated. I also also add custom x-axis ticks and give the option Appearance -> "Labeled" to the control variable a to make the manipulation more user friendly.
Manipulate[ Plot[FourierTrigSeries[-x^2 + a x, x, 10], {x, -2 Pi, 2 Pi}, Ticks -> {Subdivide[-2 Pi, 2 Pi, 8], Automatic}, Evaluated -> True], {a, -10, 10, .2, Appearance -> "Labeled"}]
J.M.'s advice, given in his comment, is one solution. Here is another using the plot option Evaluated. I also add custom x-axis ticks and give the option Appearance -> "Labeled" to the control variable a to make the manipulation more user friendly.
Manipulate[ Plot[FourierTrigSeries[-x^2 + a x, x, 10], {x, -2 Pi, 2 Pi}, Ticks -> {Subdivide[-2 Pi, 2 Pi, 8], Automatic}, Evaluated -> True], {a, -10, 10, .2, Appearance -> "Labeled"}]
J.M.'s advice, given in his comment, is one solution. Here is another using the plot option Evaluated. I also also add custom x-axis ticks and give the option Appearance -> "Labeled" to the control variable a to make the manipulation more user friendly.
Manipulate[ Plot[FourierTrigSeries[-x^2 + a x, x, 10], {x, -2 Pi, 2 Pi}, Ticks -> {Subdivide[-2 Pi, 2 Pi, 8], Automatic}, Evaluated -> True], {a, -10, 10, .2, Appearance -> "Labeled"}]