To find the product of the polynomials \( (5x - 2) \) and \( (x - 3) \), we can use the distributive property (also known as the FOIL method for binomials). We multiply each term in the first polynomial by each term in the second polynomial.
Here are the steps:
-
Multiply \( 5x \) by \( x \): \[ 5x \cdot x = 5x^2 \]
-
Multiply \( 5x \) by \( -3 \): \[ 5x \cdot (-3) = -15x \]
-
Multiply \( -2 \) by \( x \): \[ -2 \cdot x = -2x \]
-
Multiply \( -2 \) by \( -3 \): \[ -2 \cdot (-3) = 6 \]
Now, we will combine all these results: \[ 5x^2 - 15x - 2x + 6 \]
Next, we combine the like terms (\(-15x\) and \(-2x\)): \[ 5x^2 - 17x + 6 \]
Thus, the product of the polynomials \( (5x - 2) \) and \( (x - 3) \) is: \[ \boxed{5x^2 - 17x + 6} \]