# Question Video: Using an Area Model to Multiply Polynomials Mathematics

Which of the following equations is represented by the given area model? [A] (𝑎 + 5)(𝑎 − 3) = 𝑎² + 17𝑎 − 15 [B] (𝑎 + 5)(4𝑎 − 3) = 4𝑎² + 17𝑎 − 15 [C] (𝑎 + 5)(4𝑎 + 3) = 4𝑎² − 17𝑎 − 15 [D] (4𝑎 + 5)(𝑎 − 3) = 4𝑎² − 7𝑎 − 15 [E] (4𝑎 − 5)(𝑎 + 3) = 4𝑎² + 7𝑎 − 15

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### Video Transcript

Which of the following equations is represented by the given area model? Is it (A) 𝑎 plus five multiplied by 𝑎 minus three is equal to 𝑎 squared plus 17𝑎 minus 15? (B) 𝑎 plus five multiplied by four 𝑎 minus three is equal to four 𝑎 squared plus 17𝑎 minus 15. (C) 𝑎 plus five multiplied by four 𝑎 plus three is equal to four 𝑎 squared minus 17𝑎 minus 15. (D) Four 𝑎 plus five multiplied by 𝑎 minus three is equal to four 𝑎 squared minus seven 𝑎 minus 15. Or (E) four 𝑎 minus five multiplied by 𝑎 plus three is equal to four 𝑎 squared plus seven 𝑎 minus 15.

We know that in order to calculate the area of any rectangle, we multiply its length by its width. This means that when we consider the top-left rectangle, 𝑎 multiplied by something is equal to four 𝑎 squared. To work out the missing length, we can divide both sides of this equation by 𝑎. When dividing four 𝑎 squared by 𝑎, we can cancel an 𝑎. This leaves us with four 𝑎. The first missing term is therefore four 𝑎.

We can repeat this process to work out the missing term in the top-right corner. This time, 𝑎 multiplied by something is equal to negative three 𝑎. Dividing both sides by 𝑎, once again, gives us negative three. 𝑎 multiplied by negative three is negative three 𝑎. The final missing term on the outside of our area model can be calculated using the equation four 𝑎 multiplied by something is equal to 20𝑎. Dividing 20𝑎 by four 𝑎 gives us five. The three missing values on the outside are four 𝑎, negative three, and five.

We can then calculate the missing value inside the area model by multiplying five and negative three. Multiplying a positive number by a negative number gives a negative answer. So, five multiplied by negative three is negative 15. One of the dimensions of our rectangle is therefore 𝑎 plus five. Our other dimension is four 𝑎 minus three. We can calculate the area of the entire rectangle by multiplying 𝑎 plus five by four 𝑎 minus three. The only option that corresponds to this is option (B) which suggests this is the correct answer.

Inside our area model, we have four 𝑎 squared minus three 𝑎 plus 20𝑎 minus 15. Our middle two terms simplify to 17𝑎 as negative three plus 20 is 17. 𝑎 plus five multiplied by four 𝑎 minus three is equal to four 𝑎 squared plus 17𝑎 minus 15. This confirms that the correct answer is option (B). This is the equation that is represented by the area model.