Question

multiply and simplify, if necessary:

\\(6a + 5)(a - 6) + (a - 3)(a - 3)\\

Explanation:

Expand the first product

We begin by multiplying the first two binomials. We will apply the FOIL Method to expand this expression.

$$ (6a + 5)(a - 6) = (6a)(a) + (6a)(-6) + (5)(a) + (5)(-6) $$

$$ = 6a^2 - 36a + 5a - 30 $$

$$ = 6a^2 - 31a - 30 $$

Expand the second product

Next, we expand the second part of the expression. This is a binomial squared, so we can apply the Square of a Binomial Difference formula.

$$ (a - 3)(a - 3) = (a - 3)^2 $$

$$ = a^2 - 2(a)(3) + 3^2 $$

$$ = a^2 - 6a + 9 $$

Combine the expanded parts

Now, we write the full expression by adding the two expanded polynomial parts together, demonstrating Polynomial Addition.

$$ (6a^2 - 31a - 30) + (a^2 - 6a + 9) $$

Group and combine like terms

To perform Polynomial Simplification, we group the terms with the same powers of \(a\) together and simplify.

$$ = (6a^2 + a^2) + (-31a - 6a) + (-30 + 9) $$

$$ = 7a^2 - 37a - 21 $$

Answer:

Multiply and simplify, if necessary:

$$(6a + 5)(a - 6) + (a - 3)(a - 3) =$$

<blank>\(7a^2 - 37a - 21\)</blank>