Step 1: The picture shows atopic dermatitis (eczema), a chronic inflammation of the skin producing itchy, red, swollen and cracked skin.
Step 2: It is also called atopic eczema, is long-lasting, and tends to flare periodically. It is common in children and usually begins in infancy or early childhood.
Step 3: A family history of atopy is the key clue here - the question stem describes a child with an asthmatic mother. Children with a family history of atopic dermatitis, asthma or hay fever are more likely to develop the disease.
Step 4: SLE (b) gives a photosensitive malar rash, erythema (c) is only non-specific redness, and TEN (d) is an acute life-threatening drug reaction with widespread skin sloughing - none fits a chronic itchy eczema in an atopic child.