My way of doing this is similar to traditional long division. The divisor goes to the left of the division bracket and the dividend goes inside the division bracket. When writing a binomial, every degree from the degree of the highest term to degree zero must be represented in a term. This means that is there is a degree in the polynomial that is not represented in a term, it must be filled in by writing a term in that degree with a coefficient of zero. So in this example, t2 - 13 should be written as t2 + 0t - 13. We need to do this because when we do the algorithm, we'll want all terms of the same kind to be perfectly lined up in columns all the way down to the finish.
So, to the left of the division bracket, write t + 4. Inside the division bracket, write t2 + 0t - 13.
Divide the first term of the dividend (t2) by the first term of the divisor (t). The answer is t. Write the answer as the first term on top of the division bracket.
Multiply the answer (t) by the divisor (t+4). The answer is t2 + 4t. Write the answer under the dividend making sure that the terms of like degrees line up vertically. That means to write the t2 from the answer under the t2 in the dividend and write the +4t from the answer under the +0t of the dividend. Draw a horizontal line under the answer.
Since we wrote two terms as the answer, we will subtract them from the two terms above. So subtract t2 + 4t from t2 + 0t. The answer is -4t. Write the answer under the horizontal line making sure again to line up like terms.
Let's introduce the next term from the divisor to the algorithm below. Bring down the -13 from the divisor and write it to the right of the -4t from the answer above. We now have a new dividend of -4t - 13.
You should start to see something familiar emerge. The old algorithm of divide, multiply, subtract, bring down, and repeat you remember from elementary long division should be obvious here.
Next, we will divide the first term of the new dividend (-4t) by the first term of the divisor (t). The answer is -4. Write the answer on top of the division bracket to the right of the t that we got from the last time we divided. The quotient should now say t - 4. This is the completed quotient, but we must press on to find the remainder.
Multiply the -4 from the quotient on by the divisor (t + 4). The answer is -4t - 16. Write this under the new dividend (-4t - 13) making sure to line up like terms. draw a horizontal line under the answer.
Subtract the answer (-4t - 16) from the new dividend (-4t - 13). The answer is 3. This is the remainder. Write the remainder under the horizontal line.
So (t2 -13) / (t + 4) = t - 4 with a remainder of 3.