Step 1 :The area of a rectangle is given by the formula length * width. In this case, the length and width of the picture (not including the mat and frame) would be \((40-2x)\) and \((19-2x)\) respectively, because we subtract the width of the mat and frame (which is x on each side) from the total length and width.
Step 2 :Therefore, the equation that represents the area of the picture itself being 700 square inches would be \((40-2x)(19-2x) = 700\).
Step 3 :Solving this equation gives us two possible solutions for x: \(\frac{59}{4} - \frac{\sqrt{3241}}{4}\) and \(\frac{\sqrt{3241}}{4} + \frac{59}{4}\).
Step 4 :Final Answer: The correct answer is \(\boxed{\text{(A) } (40-2x)(19-2x) = 700}\).