To solve the linear equation $0.25p + 0.25(p + 4) = 5.20$, begin by applying the distributive property to the left side, yielding $0.25p + 0.25p + 1 = 5.20$. Next, combine the like decimal variable terms to obtain $0.50p + 1 = 5.20$. Isolate the variable term by subtracting $1$ from both sides, which results in $0.50p = 4.20$. Finally, divide both sides by the coefficient $0.50$ to find the solution, $p = 8.4$. Another valid approach is to clear the decimals immediately by multiplying every term in the equation by $100$, producing $25p + 25(p + 4) = 520$, which will give the same result.