To multiply the binomials [tex]\((3x - 5)(-x + 4)\)[/tex] and simplify the product into standard form, we'll follow the distributive property (also known as the FOIL method for binomials, which stands for First, Outer, Inner, Last).
Here's the step-by-step process:
1. First: Multiply the first terms in each binomial.
[tex]\[
(3x) \cdot (-x) = -3x^2
\][/tex]
2. Outer: Multiply the outer terms.
[tex]\[
(3x) \cdot 4 = 12x
\][/tex]
3. Inner: Multiply the inner terms.
[tex]\[
(-5) \cdot (-x) = 5x
\][/tex]
4. Last: Multiply the last terms in each binomial.
[tex]\[
(-5) \cdot 4 = -20
\][/tex]
Next, we combine all these products together:
[tex]\[
-3x^2 + 12x + 5x - 20
\][/tex]
Now, combine like terms:
[tex]\[
-3x^2 + (12x + 5x) - 20 = -3x^2 + 17x - 20
\][/tex]
So, the simplified product in standard form is:
[tex]\[
-3x^2 + 17x - 20
\][/tex]
Thus, the expression can be written as:
[tex]\[
-3x^2 + 17x - 20
\][/tex]
